Landscapes of Time-Frequency Analysis - ATFA 2019 [1 ed.] 9783030560041

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Applied and Numerical Harmonic Analysis

Paolo Boggiatto · Tommaso Bruno Elena Cordero · Hans G. Feichtinger Fabio Nicola · Alessandro Oliaro Anita Tabacco · Maria Vallarino Editors

Landscapes of Time-Frequency Analysis ATFA 2019

Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA

Advisory Editors Akram Aldroubi Vanderbilt University Nashville, TN, USA

Gitta Kutyniok Technical University of Berlin Berlin, Germany

Douglas Cochran Arizona State University Phoenix, AZ, USA

Mauro Maggioni Johns Hopkins University Baltimore, MD, USA

Hans G. Feichtinger University of Vienna Vienna, Austria

Zuowei Shen National University of Singapore Singapore, Singapore

Christopher Heil Georgia Institute of Technology Atlanta, GA, USA

Thomas Strohmer University of California Davis, CA, USA

Stéphane Jaffard University of Paris XII Paris, France

Yang Wang Hong Kong University of Science & Technology Kowloon, Hong Kong

Jelena Kovaˇcevi´c Carnegie Mellon University Pittsburgh, PA, USA

More information about this series at http://www.springer.com/series/4968

Paolo Boggiatto • Tommaso Bruno • Elena Cordero Hans G. Feichtinger • Fabio Nicola • Alessandro Oliaro • Anita Tabacco • Maria Vallarino Editors

Landscapes of Time-Frequency Analysis ATFA 2019

Editors Paolo Boggiatto Department of Mathematics University of Turin Torino, Italy

Tommaso Bruno Department of Mathematics Ghent University Ghent, Belgium

Elena Cordero Department of Mathematics University of Turin Torino, Italy

Hans G. Feichtinger Institute of Mathematics University of Vienna Wien, Austria

Fabio Nicola Department of Mathematical Sciences Polytechnic University of Turin Torino, Italy

Alessandro Oliaro Department of Mathematics University of Turin Torino, Italy

Anita Tabacco Department of Mathematical Sciences Polytechnic University of Turin Torino, Italy

Maria Vallarino Department of Mathematical Sciences Polytechnic University of Turin Torino, Italy

ISSN 2296-5009 ISSN 2296-5017 (electronic) Applied and Numerical Harmonic Analysis ISBN 978-3-030-56004-1 ISBN 978-3-030-56005-8 (eBook) https://doi.org/10.1007/978-3-030-56005-8 Mathematics Subject Classification: 42B35, 42C15, 43A32, 44A12, 46F12, 47G10, 42C40, 65Txx, 81S30, 92C55 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to our Families, their continuous support is our strength

ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-theart ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. vii

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ANHA Series Preface

Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in the following applicable topics in which harmonic analysis plays a substantial role: Antenna theory Biomedical signal processing Digital signal processing Fast algorithms Gabor theory and applications Image processing Numerical partial differential equations

Prediction theory Radar applications Sampling theory Spectral estimation Speech processing Time-frequency and Time-scale analysis Wavelet theory

The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function.” Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or the

ANHA Series Preface

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adaptivemodeling inherent in time-frequency-scalemethods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’être of the ANHA series! University of Maryland College Park

John J. Benedetto Series Editor

Preface

The second international conference entitled “Aspects of Time-Frequency Analysis (ATFA19)” took place in the period of 25–27 June 2019, at DISMA, Politecnico di Torino, see http://www.atfa19.polito.it/. The local organizing committee consisted of people from both university institutions of the city: Tommaso Bruno, Fabio Nicola, Anita Tabacco, and Maria Vallarino (Politecnico di Torino) and Paolo Boggiatto, Elena Cordero, and Alessandro Oliaro (Università di Torino). The present volume collects ten contributions, mostly from invited speakers at the conference. These articles cover a good selection of topics of current interest in the field. They appear in alphabetic order of the first author, but let us review them following the topics they address in this short summary. Let me start with those contributions that connect the area of time-frequency analysis with real-world applications. The first article to be mentioned here is the contribution by Leon Cohen, entitled “Time-Frequency Analysis: What We Know and What We Don’t”. It is a great opportunity to hear from the author of a wellknown book in the area [2] about his current view on the field. In fact, there is a surprising little overlap of the list of references in his book, as well as in the book of B. Boashash [1] with the same title, with the standard references in the mathematical literature, specifically [4] and [3]. This indicates that there is more need to establish better connections and interactions between engineers and mathematicians. The connection to physics plays an important role in the contributions of J.P. Gazeau and C. Habonimana, entitled “Signal Analysis and Quantum Formalism: Quantizations with No Planck Constant” and M. de Gosson’s “Generalized AntiWick Quantum States”. Both establish the connection to quantum theory, which from the very beginning was one of the motivations of time-frequency analysis, in terms of the well-known coherent states. The article by S.I. Trapasso “A Time-Frequency Analysis Perspective on Feynman Path Integrals” shows how to make use of the modulation space theory to establish rigorous mathematical statements inspired by the ideas of R. Feynman concerning path integrals. However, one finds in the article written by D. Labate, B.R. Pahari, S. Hoteit, and M. Mecati “Quantitative Methods in Ocular Fundus Imaging: Analysis of Retinal Microvasculature” how time-frequency analysis methods find xi

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a concrete application in the medical sciences, via the example of the anatomic structure of the retina. This article also describes practical issues of data handling or the segmentation of retinal vessels. A somewhat more abstract viewpoint, still related to data handling, is provided by the article “Data Approximation with Time-Frequency Invariant Systems” by D. Barbieri, C. Cabrelli, E. Hernández, and U. Molter. It is explained in the article how an optimal system (within a given TF framework) can be determined, for a given data set. The contribution by F. Bartolucci, S. Pilipovi´c, and N. Teofanov on “The Shearlet Transform and Lizorkin Spaces” describes specific aspects of the mathematical foundation of shearlet theory, namely the correct choice of the space of test functions to be used, connecting modern shearlet theory with a construction going back to Lizorkin (in his studies of inhomogeneous Besov spaces, if we use modern terminology), also related to the setting of Triebel–Lizorkin spaces in the terminology of H. Triebel’s universe of the “Theory of Function Spaces” [6–8]. There is also a strong group theoretic connection shown in the opening article “Radon Transform: Dual Pairs and Irreducible Representations” by S. Alberti, F. Bartolucci, F. De Mari, and E. De Vito. This contribution connects in a very interesting way wavelet transform theory with group representations in the spirit of S. Helgason (see, e.g., [5]). Finally, let us mention two further articles related to anti-Wick (respectively, Toeplitz) operators in the time-frequency context: F. Bastianoni reports in “TimeFrequency Localization Operators: State of the Art”, while R. Corso and F. Tschinke report in “Some Notes About Distribution Frame Multipliers”. They treat different aspects of operators, which are realized as multiplication operators on the transform domain. While the first one covers the case of the STFT (short-time Fourier transform) and results in anti-Wick-type operators, the second concentrates on a theme, again close to mathematical physics, where one has a continuous family of distributions forming a kind of continuous basis. Here, the family (δx )x∈R or the family of pure frequencies are prototypical examples. All those who have contributed to this volume are grateful to the Proceedings team and to the publisher (Birkhäuser) for organizing this book in their prestigious ANHA (Applied and Numerical Harmonic Analysis) series. We acknowledge their patience and perseverance in keep reminding some of us, to finally come up with this interesting collection of articles. Given the fact that this was already the second conference in this series, the participants (and those who missed the opportunity of joining us this time) may hope for another event like this in Torino in the years to come, even though Italy is hit by a pandemic crisis at the time of writing of this preface. Thus, we also wish Italy and especially our colleagues in Torino a good and prompt recovery. We are looking forward to visit Italy in the not too distant future and enjoy another topical conference there. Wien, Austria 27 March 2020

Hans G. Feichtinger

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References 1. B. Boashash, editor. Time-Frequency Analysis. Methods and Applications. Longman Cheshire and Halsted Press, Wiley, 1992. 2. L. Cohen. Time-Frequency Analysis: Theory and Applications. Prentice Hall Signal Processing Series, Prentice Hall, 1995. 3. G. B. Folland. Harmonic Analysis in Phase Space. Princeton University Press, Princeton, N.J., 1989. 4. K. Gröchenig. Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal. Birkhäuser, Boston, MA, 2001. 5. S. Helgason. The Radon Transform. Progress in Mathematics, Boston, MA, Birkhäuser, 1999. 6. H. Triebel. Theory of Function Spaces. Vol. 78 of Monographs in Mathematics. Birkhäuser, Basel, 1983. 7. H. Triebel. Theory of Function Spaces II. Vol. 84 of Monographs in Mathematics. Birkhäuser, Basel, 1992. 8. H. Triebel. Theory of Function Spaces III, Vol. 100 of Monographs in Mathematics. Birkhäuser, Basel, 2006.

Acknowledgements

The editors recognize the indispensable financial support of the following institutions: 1. Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (partially MIUR grant Dipartimenti di Eccellenza 20182022, CUP: E11G18000350001, DISMA, Politecnico di Torino). 2. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy. Due to the funds from the aforementioned institutions, the organizers were able to cover travel and living expenses of many participants, including numerous young researchers, graduate students, and postdocs. They are also very grateful to the NuHAG (Numerical Harmonic Analysis Group), Faculty of Mathematics, University of Vienna, for the longstanding scientific collaboration and friendship and for contributing to make ATFA19 (Aspects of Time-Frequency Analysis) so stimulating and fruitful.

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Contents

Radon Transform: Dual Pairs and Irreducible Representations . . . . . . . . . . . Giovanni S. Alberti, Francesca Bartolucci, Filippo De Mari, and Ernesto De Vito 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Setting and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Unitarization Theorem and Inversion Formula . . . . . . . . . . . . . . . . . . . 3 Dual Pairs and Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Irreducibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3D-Signals: Radon and Ray Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Radon Transform on R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The X-ray Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1 4 4 5 8 9 11 17 17 23 27

Data Approximation with Time-Frequency Invariant Systems . . . . . . . . . . . . . Davide Barbieri, Carlos Cabrelli, Eugenio Hernández, and Ursula Molter 1 Introduction and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 An Isometric Isomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solution to the Approximation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Shearlet Transform and Lizorkin Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Francesca Bartolucci, Stevan Pilipovi´c, and Nenad Teofanov 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Affine Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Shearlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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29 32 38 41 42

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4 The Shearlet Synthesis Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Shearlet Transform on S0 (R2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 58 60 62

Time–Frequency Localization Operators: State of the Art . . . . . . . . . . . . . . . . . Federico Bastianoni 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Boundedness, Compactness and Schatten-von Neumann Class . . . . . . . . . . . . 2.1 Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Main Results on L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Gelfand–Shilov Spaces Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Gelfand–Shilov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Ultra-Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Boundedness and Schatten Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Time-Frequency Analysis: What We Know and What We Don’t . . . . . . . . . . Leon Cohen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Why the Absolute Square of the Function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Why Do We Need a Time-Varying Spectrum Theory?. . . . . . . . . . . . . . . . 2 The Joint Density Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Arguments Against a Proper Joint Density Theory . . . . . . . . . . . . . . . . . . . . . 3.1 Uncertainty Principle Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Impossibility Theorem of Wigner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 One Cannot Define Frequency at a Point in Time . . . . . . . . . . . . . . . . . . . . . 3.4 Quantum Mechanics Has Shown that One Cannot Measure Physical Quantities to Arbitrary Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Uncertainty Product Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Time and Frequency Resolution “Trade Off” . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The Uncertainty Principle for the Spectrogram. . . . . . . . . . . . . . . . . . . . . . . . 4 Classification of Time-Frequency Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Bilinear Distributions Satisfying the Marginals . . . . . . . . . . . . . . . . . . . . . . . 4.2 Manifestly Positive Bilinear Densities Not Satisfying the Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Non-bilinear, Manifestly Positive Distributions Satisfying the Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Why Is There an Exception to Wigner’s Theorem? . . . . . . . . . . . . . . . . . . . . . . . . . 6 Is Time-Frequency “Hidden” in Standard Fourier Analysis? . . . . . . . . . . . . . . . 7 Can One Derive Time-Frequency Densities? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 A New Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 What Did Wigner Do in His 1932 Paper? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Wigner Distribution as a Tool for Solving Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 65 65 68 69 70 71 71 72 73 73 75 75 76 76 77 78 78 81 81 81 81 82 83 84 84 85 86 88 90 91 93 94 95 98

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Some Notes About Distribution Frame Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . Rosario Corso and Francesco Tschinke 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminary Definitions and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Distribution Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Unbounded Distribution Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Riesz Distribution Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Generalized Anti-Wick Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maurice de Gosson 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Density Operators: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Toeplitz Operators: Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Toeplitz Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Discussion and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Signal Analysis and Quantum Formalism: Quantizations with No Planck Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean Pierre Gazeau and Célestin Habonimana 1 Introduction: The Planck Constant h¯ Is for What? . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fourier, Gabor, and Wavelet Analysis in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fourier Analysis, Like in Euclidean Geometry. . . . . . . . . . . . . . . . . . . . . . . 2.2 Gabor Signal Analysis (∼ Time-Frequency) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Continuous Wavelet Transform (∼ Time-Scale) . . . . . . . . . . . . . . . . . . . . . . 3 From Signal Analysis to Quantum Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Resolution of the Identity as the Common Guideline . . . . . . . . . . . . . . . . . 3.2 Probabilistic Content of Integral Quantization: Semi-Classical Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 First Examples of Operators M(x): Projector-Valued (PV) Measures for Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 PV Measures for Quantization: Time Operator. . . . . . . . . . . . . . . . . . . . . . . . 4.2 PV Measures for Quantization: Frequency Operator . . . . . . . . . . . . . . . . . . 4.3 Mutatis Mutandis. . . and CCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Limitations of PV Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Weyl–Heisenberg Covariant Integral Quantization with Gabor and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 From PV Quantization to Gabor POVM Quantization . . . . . . . . . . . . . . . . 5.2 Beyond Gabor Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Affine Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 105 111 114 118 120 121

123 123 125 126 128 130 133 134 135 135 138 138 139 140 141 141 141 142 143 143 144 144 145 145 145 146 149

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7

Examples of Operatorial Signal Analysis Through Gabor Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Quantitative Methods in Ocular Fundus Imaging: Analysis of Retinal Microvasculature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Demetrio Labate, Basanta R. Pahari, Sabrine Hoteit, and Mariachiara Mecati 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Anatomy of the Retina and Retinal Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Anatomic Structure of the Retina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Retinal Manifestations of Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fundus Photography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Image Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Automated Image Analysis of Retinal Vascularization. . . . . . . . . . . . . . . . . . . . . . 3.1 Local Measures of Retinal Vascularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Global Measures of Retinal Vascularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Segmentation of Retinal Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A Time–Frequency Analysis Perspective on Feynman Path Integrals . . . . . S. Ivan Trapasso 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A Few Facts on Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Two Rigorous Approaches to Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Sequential Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Time-Slicing Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Beyond the L2 Theory via Gabor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Role of Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Sparse Operators on Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 An Unavoidable Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Higher Order Rough Parametrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Role of h¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Role of M ∞,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Pointwise Convergence of Integral Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Weyl Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Proof at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Why Exceptional Times? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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157 159 159 160 161 161 162 163 164 167 169 175 175 178 180 180 182 185 186 187 189 189 192 193 193 194 195 198 199 200

Contributors

Giovanni S. Alberti Department of Mathematics and MaLGa Center, University of Genoa, Genova, Italy Davide Barbieri Universidad Autónoma de Madrid, Madrid, Spain Francesca Bartolucci Department of Mathematics, ETH Zurich, Zurich, Switzerland Federico Bastianoni Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Torino, Italy Carlos Cabrelli Departamento de Matemática, Universidad de Buenos Aires, and Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina Leon Cohen Hunter College of The City University of New York, New York, NY, USA Rosario Corso Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy Maurice de Gosson Faculty of Mathematics, NuHAG, University of Vienna, Wien, Austria Institute of Mathematics, University of Würzburg, Würzburg, Germany Filippo De Mari Department of Mathematics and MaLGa Center, University of Genoa, Genova, Italy Ernesto De Vito Department of Mathematics and MaLGa Center, University of Genoa, Genova, Italy Jean Pierre Gazeau Astroparticules et Cosmologie, Université Paris Diderot, Paris, France Célestin Habonimana Université du Burundi and École Normale Supérieure, Bujumbura, Burundi xxi

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Eugenio Hernández Universidad Autónoma de Madrid, Madrid, Spain Sabrine Hoteit Department of Mathematics, University of Houston, Houston, TX, USA Demetrio Labate Department of Mathematics, University of Houston, Houston, TX, USA Mariachiara Mecati Dipartimento di Automatica e Informatica, Politecnico di Torino, Torino, Italy Ursula Molter Departamento de Matemática, Universidad de Buenos Aires, and Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina Basanta R. Pahari Department of Mathematics, University of Houston, Houston, TX, USA Stevan Pilipovi´c Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia Nenad Teofanov Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia S. Ivan Trapasso Dipartimento di Scienze Matematiche (DISMA) “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Francesco Tschinke Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy

Radon Transform: Dual Pairs and Irreducible Representations Giovanni S. Alberti, Francesca Bartolucci, Filippo De Mari, and Ernesto De Vito

Abstract We illustrate the general point of view we developed in an earlier paper (SIAM J. Math. Anal., 2019) that can be described as a variation of Helgason’s theory of dual G-homogeneous pairs (X, Ξ ) and which allows us to prove intertwining properties and inversion formulae of many existing Radon transforms. Here we analyze in detail one of the important aspects in the theory of dual pairs, namely the injectivity of the map label-to-manifold ξ → ξˆ and we prove that it is a necessary condition for the irreducibility of the quasi-regular representation of G on L2 (Ξ ). We further explain how our construction applies to the classical Radon and X-ray transforms in R3 . Keywords Homogeneous spaces · Radon transform · Dual pairs · Square-integrable representations · Inversion formula · Wavelets · Shearlets

1 Introduction The circle of ideas and problems that may be collectively named “Radon transform theory” was born at least a century ago [17] but still abounds with questions and new perspectives that range from very concrete computation-oriented tasks to geometric or representation theoretic issues. We may describe the heart of the matter by paraphrasing Gelfand [8]: “Let X be some space and in it let there be given certain manifolds which we shall suppose to be analytic and dependent analytically on parameters ξ1 , . . . , ξk , that is {ξˆ (ξ ) = ξˆ (ξ1 , . . . , ξk )}. With a function f on X we associate its integrals over these manifolds: G. S. Alberti · F. De Mari · E. De Vito () Department of Mathematics and MaLGa Center, University of Genoa, Genova, Italy e-mail: [email protected]; [email protected]; [email protected] F. Bartolucci Department of Mathematics, ETH Zurich, Zurich, Switzerland e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_1

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 Rf (ξ ) =

f (x)dmξ (x). ξˆ

We then ask whether it is possible to determine f knowing the integrals Rf (ξ ).” Among the many generalizations and theorems that may be subsumed in this basic, yet profound, mathematical sketch, it is certainly worth mentioning Helgason’s contribution, inspired [12] by work of Fritz John’s, in turn triggered by Radon’s original result [17] dating back to 1917. In particular, Helgason developed the notion of dual pairs and double fibrations, whereby (Lie) groups and homogeneous spaces thereof stand at center stage. His basic observation comes by inspecting John’s inversion formula for the integral transform–nowadays the prototypical Radon transform–defined by integration over planes in R3 . The inversion takes the form   1 f (x) = − 2 Δx Rf (n, n · x)dn , 8π S2

where (n, t) → Rf (n, t) is the function on S 2 × R given by the integral of f over the plane ξˆ (n, t) = {x ∈ R3 : n · x = t}, Δx is the Laplacian, and dn is the Riemannian measure on the sphere S 2 . This formula, observes Helgason [12], “involves two dual integrations, Rf is the integral over the set of points in a plane and then dn, the integral over the set of planes through a point.” Furthermore, the domain X on which the functions of interest are defined (here X = R3 ) and the set Ξ of relevant manifolds (here the two-dimensional planes) are homogeneous spaces of the same group G, namely the group of isometries of R3 , and enjoy a sort of duality, well captured by the differential-geometric notion of incidence that was introduced by Chern [5]. Helgason proceeds on developing this duality in group-theoretic terms, emphasizing a remarkable formal symmetry, according to which the objects of interest come naturally in pairs, one living in X and its twin in Ξ . Most notably, each point ξ ∈ Ξ (the pair (ξ1 , ξ2 ) = (n, t) in our basic example) labels one of the actual submanifolds ξˆ of X on which the relevant integrals are to be taken (the plane ξˆ (n, t)). Conversely, with each point x ∈ X it is natural to associate the “sheaf ” of planes passing through it. In the example above, this is precisely the set xˇ = {ξˆ (n, x · n) : n ∈ S 2 } over which the integral of Rf is taken. In the abstract setting developed by Helgason, the whole construction enjoys natural properties as long as the mappings ξ → ξˆ and x → xˇ are both injective, requirement that is then built into the definition of dual pair and expressed algebraically. Note that in the above example, the map (n, t) → ξˆ (n, t) is two-toone and this lack of injectivity is reflected by the fact that Rf is an even function. The central object is of course the Radon transform  Rf (ξ ) = f (x) dmξ (x) ξˆ

for integrable functions on X, where mξ is a suitable measure on ξˆ .

Radon Transform: Dual Pairs and Irreducible Representations

3

Utilizing a variation of this framework, which is recalled in full detail below, we have addressed [1] some issues that are naturally expressed in this language. Our main contribution (see Theorem 1) is a general result concerning the “unitarization” of R from L2 (X, dx) to L2 (Ξ, dξ ) and the fact that the resulting unitary operator intertwines the quasi-regular representations π and πˆ of G on L2 (X, dx) and L2 (Ξ, dξ ), respectively. This unitarization really means first pre-composing the closure of R with a suitable pseudo-differential operator and then extending this composition to a unitary map, as is done in the existing and well-known predecessors of Theorem 1, such as those in [11] and in [21]. The representations π and πˆ play of course a central role and are assumed to be irreducible, and π is assumed to be square-integrable (see Assumptions (A4) and (A5) below). The combination of unitary extension and intertwining leads to an interesting inversion formula for the true Radon transform, see Theorem 2. Compared to [1], the present article adopts a slightly different, though fully compatible, formalism in the sense that we take here the point of view that seems most natural in applications. Indeed, the space X where the signals of interest are defined and the set of submanifolds of X where integrals are to be taken are both in the foreground, and the group G of geometric actions that one wants to consider comes next, tailored to the problem at hand. In this regard, it is important to observe that, in principle, there are many different realizations of X as homogeneous space, and the choice of G is tantamount to choosing the particular set of transformations (or symmetries) that one wants to focus on. In this context, it is of course important that there are sufficiently many of these transformations. As for the submanifolds, we observe that in most applications one has in mind a prototypical submanifold ξˆ0 . We thus choose and fix ξˆ0 , which we refer to as the root submanifold, as the image of the base point x0 ∈ X under the action of some closed subgroup H of G. Thus ξˆ0 = H [x0 ], and the other submanifolds are obtained by exploiting the fact that X is a transitive G-space. This entails that X is covered with all the shifted versions of ξˆ0 by means of the geometric transformations given by the elements of G. Incidentally, in this way one often achieves families of foliations, and in most cases this leads to a natural splitting of the parameters in Ξ , those that label the foliation and those that select the leaf in the foliation. Although largely inspired by the work of Helgason, our approach is different in several ways that are discussed in detail in Sect. 2. His construction rests not only on the strict invariance of the measures on X, Ξ , and ξˆ0 (versus relative invariance as in our construction) but also on the fact that the correspondence ξ → ξˆ between “labels” in the transitive G-space Ξ and submanifolds of X is assumed to be injective. In the present article we investigate this issue in detail and focus on the  of G that fixes ξˆ0 , in principle larger than H . We find (Proposition 1) subgroup H  = H and we further show in that the map ξ → ξˆ is injective if and only if H , if this equality fails, then πˆ Theorem 3 that, under reasonable assumptions on H cannot be irreducible. This implies that in order for Assumption (A5) to be fulfilled, one must choose H as large as possible among those subgroups of G that fill out ξˆ0

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by acting on x0 . Our theory is then illustrated with the help of two examples, namely the classical Radon transform and the X-ray transform in R3 , both analyzed with the group SIM(3) of rotations, dilations, and translations. Again, this is different from Helgason’s standard choice, the isometry group M(3). The paper is organized as follows. In Sect. 2 we set up the context and recall the main results of [1]. In Sect. 3 we present a rather detailed analysis of the relations existing between the objects naturally arising from an arbitrary choice of H and . This leads to the main contribution those that come from the maximal choice H  and H implies that the quasiof this work, namely the fact that a gap between H 2 regular representation πˆ of G on L (Ξ ) cannot be irreducible. Section 4 illustrates our theory with two classical examples in three-dimensional Euclidean space.

2 The Framework In this section we introduce the setting and the main result of [1].

2.1 Notation We briefly introduce the notation. We set R× = R \ {0} and R+ = (0, +∞). The Euclidean norm of a vector v ∈ Rd is denoted by |v| and its scalar product with w ∈ Rd by v · w. For any p ∈ [1, +∞] we denote by Lp (Rd ) the Banach space of functions f : Rd → C that are p-integrable with respect to the Lebesgue measure dx and, if p = 2, the corresponding scalar product and norm are ·, · and · , respectively. If E is a Borel subset of Rd , |E| denotes its Lebesgue measure. The Fourier transform is denoted by F both on L2 (Rd ) and on L1 (Rd ), where it is defined by  F f (ω) =

f (x) e−2π i ω·x dx,

f ∈ L1 (Rd ).

Rd

If G is a locally compact second countable (lcsc) group, we denote by L2 (G, μG ) the Hilbert space of square-integrable functions with respect to a left Haar measure μG on G. If X is a lcsc transitive G-space with origin x0 , we denote by g[x] the action of G on X. A Borel measure ν on X is relatively invariant if there exists a positive character α of G such that for any measurable set E ⊆ X and g ∈ G it holds ν(g[E]) = α(g)ν(E), see, e.g., [19]. Furthermore, a Borel section is a measurable map s : X → G satisfying s(x)[x0 ] = x and s(x0 ) = e, with e the neutral element of G; a Borel section always exists since G is second countable [22, Theorem 5.11]. We denote the (real) general linear group of size d × d by GL(d, R).

Radon Transform: Dual Pairs and Irreducible Representations

5

Given two unitary representations π, πˆ of G acting on two Hilbert spaces H and Hˆ , respectively, a densely defined closed operator T : H → Hˆ is called semi-invariant with weight ζ if it satisfies πˆ (g) T π(g)−1 = ζ (g)T ,

g ∈ G,

(1)

where ζ is a character of G, see [6].

2.2 Setting and Assumptions The Radon transform of a signal f : X → C is defined as the integral of f over a suitable family {ξˆ } of subsets of X indexed by a label ξ ∈ Ξ . In this paper, we assume that the input space X is a lcsc space and the signals are elements of the Hilbert space L2 (X, dx), where dx is a given measure on X, defined on the Borel σ -algebra of X and finite on compact subsets. Following Helgason’s approach, the family {ξˆ } is defined by first choosing a lcsc group G acting on X by a continuous action (g, x) → g[x]

(2)

in such a way that X becomes a transitive G-space. Then, we fix an origin x0 ∈ X, a closed subgroup H of G and we define the root ξˆ0 of the family {ξˆ } as ξˆ0 = H [x0 ],

(3)

which is a closed H -invariant subset of X, by [11, Lemma 1.1]. Denote the set of left cosets by Ξ = G/H and define for each ξ = gH ∈ Ξ the closed subset of X ξˆ = g[ξˆ0 ] = gH [x0 ],

(4)

which is independent of the choice of the representative g ∈ G of ξ ∈ G/H . In order to view the roles played by X and Ξ as somewhat symmetric, we introduce the stability subgroup of G at x0 K = {k ∈ G : k[x0 ] = x0 }, which is a closed subgroup of G such that X can be identified with G/K by means of the map G/K gK → g[x0 ] ∈ X. Conversely, we regard Ξ as a transitive lcsc space with respect to the continuous action of G given by g.ξ = (gg  )H,

ξ = g  H ∈ Ξ,

(5)

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and we choose, as origin, the point ξ0 = eH , which makes (3) and (4) consistent with each other (see Lemma 1 below). With this setting, we need the following conditions to hold true: (A1) the measure dx is relatively G-invariant with character α and there exists a relatively invariant measure dξ on Ξ with character β; (A2) there exists a relatively H -invariant measure m0 on ξˆ0 with character γ ; (A3) there exist a Borel section σ : Ξ → G for the action (5) and a character ι of G such that   g ∈ G, ξ ∈ Ξ ; (6) γ σ (ξ )−1 gσ (g −1 . ξ ) = ι(g), (A4) the quasi-regular representation π of G acting on L2 (X, dx) as π(g)f (x) = α(g)−1/2 f (g −1 [x]) is irreducible and square-integrable; (A5) the quasi-regular representation πˆ of G acting on L2 (Ξ, dξ ) as πˆ (g)F (ξ ) = β(g)−1/2 F (g −1 . ξ ) is irreducible; (A6) there exists a non-trivial π -invariant subspace A ⊆ L2 (X, dx) such that for all f ∈ A f (σ (ξ )[·]) ∈ L1 (ξˆ0 , m0 ) for almost all ξ ∈ Ξ,  Rf := f (σ (·)[x]) dm0 (x) ∈ L2 (Ξ, dξ ),

(7a) (7b)

ξˆ0

and the adjoint of the operator R : A → L2 (Ξ, dξ ) has non-trivial domain. We add a few comments. The assumption that the measure dx is (relatively) invariant ensures that the group G acts also on the signals by means of the unitary representation π . It is worth observing that (7a) is independent of the section σ . Indeed, if σ  is another section, by Assumption (A2) we have 

|f (σ  (ξ )[x])|dm0 (x) =

ξˆ0



|f (σ (ξ )σ (ξ )−1 σ  (ξ )[x])|dm0 (x)

ξˆ0

  = γ σ  (ξ )−1 σ (ξ ) |f (σ (ξ )[x])|dm0 (x), ξˆ0

since σ (ξ )−1 σ  (ξ ) ∈ H .

Radon Transform: Dual Pairs and Irreducible Representations

7

By means of the section σ , the family {ξˆ } is given by ξˆ = σ (ξ )[ξˆ0 ] ⊆ X,

(8)

and the map x → σ (ξ )[x] is a Borel bijection from ξˆ0 onto ξˆ , so that (7b) reads as  Rf (ξ ) = f (x) dmξ (x), (9) ξˆ

where mξ is the image measure of m0 under the above bijection. Hence for any signal f belonging to A , the map Rf is precisely the Radon transform of f . Note that A is a dense subspace of L2 (X, dx) by the irreducibility of π and this property also guarantees that the adjoint of R is uniquely defined. Given the space of signals L2 (X, dx), there are possibly many different pairs (G, H ) that give rise to the same family {ξˆ } of subsets and (essentially) to the same Radon transform R. In this paper, G is chosen in such a way that π is a squareintegrable representation, so that there exists a self-adjoint operator C : dom C ⊆ L2 (X, dx) → L2 (X, dx), 1

semi-invariant with weight Δ 2 , where Δ is the modular function of G. Hence, for all ψ ∈ dom C with Cψ = 1, the voice transform Vψ (Vψ f )(g) = f, π(g)ψ ,

g ∈ G,

is an isometry from L2 (X, dx) into L2 (G, μG ). In this case the vector ψ is called admissible and we have the weakly convergent reproducing formula  f =

(Vψ f )(g) π(g)ψ dμG (g),

(10)

G

see, for example, [7, Theorem 2.25]). Equation (10) is at the basis of our reconstruction formula (14). We stress that in Helgason’s approach, the representation π is not directly considered, and hence there is no need to require it to be either irreducible or squareintegrable. This entails a larger freedom in the choice of the group G. We recall that, since X, Ξ , and ξˆ0 are transitive spaces, there always exist quasiinvariant measures on these three spaces [19]. In Helgason’s approach, it is assumed that the measures are invariant, so that they are unique up to a constant. In this paper, we only require that dx, dξ , and m0 are relatively invariant. In particular, m0 is not uniquely given (up to a constant) and the definition of the Radon transform depends not only on the family {ξˆ } but also on the measure m0 and the section σ . Since m0 is not invariant, Assumption (A3) is needed to ensure the right covariance properties of the Radon transform and in many examples it can be easily satisfied by a suitable choice of the section σ .

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2.3 The Unitarization Theorem and Inversion Formula The isometric extension problem for the Radon transform was actually addressed and implicitly solved by Helgason in the general context of symmetric spaces, see [10, Corollary 3.11]. However, as a consequence of the intertwining properties of the Radon transform it is possible to provide an alternative proof of the following result, see [1]. Theorem 1 Under the above assumptions, 1. the Radon transform R : A → L2 (Ξ, dξ ) admits a unique closure R; 2. the closure R satisfies Rπ(g) = χ (g)−1 πˆ (g)R,

(11)

for all g ∈ G, where χ is the character given by χ (g) = α(g)1/2 β(g)−1/2 γ (gσ (g −1 .ξ0 ))−1 ;

(12)

3. there exists a unique positive self-adjoint operator I : dom(I ) ⊇ Im R → L2 (Ξ, dξ ), semi-invariant with weight ζ = χ −1 with the property that the composite operator I R extends to a unitary operator Q : L2 (X, dx) → L2 (Ξ, dξ ) intertwining π and πˆ , namely πˆ (g) Q π(g)−1 = Q,

g ∈ G.

(13)

It follows that the representations π and πˆ are equivalent, so that πˆ is squareintegrable, too. Since Q is unitary and satisfies (13) and π is square-integrable, it is possible to prove the following inversion formula for the Radon transform, [1]. Theorem 2 Let ψ ∈ L2 (X, dx) be an admissible vector for the representation π such that Qψ ∈ dom I , and set Ψ = I Qψ. Then, for any f ∈ dom R,  f =

χ (g)Rf, πˆ (g)Ψ π(g)ψ dμG (g),

(14)

G

where the integral is weakly convergent, and 

f =

χ (g)2 |Rf, πˆ (g)Ψ |2 dμ(g).

2

G

If, in addition, ψ ∈ dom R, then Ψ = I 2 Rψ.

(15)

Radon Transform: Dual Pairs and Irreducible Representations

9

Note that the datum Rf is analyzed by the family {π(g)Ψ ˆ }g∈G and the signal f is reconstructed by a different family, namely {π(g)ψ}g∈G . The idea to exploit the theory of the continuous wavelet transform to derive inversion formulae for the Radon transform is not new, we refer to [3, 13, 14, 16, 20, 23, 24]—to name a few.

3 Dual Pairs and Irreducibility In this section, we show the relation between our setting and the notion of dual pairs introduced by Helgason [11] and the connection with the assumption on the irreducibility. If we identify X and Ξ with the corresponding homogenous spaces G/K and G/H , so that ξ = g1 H for some g1 ∈ G, then a point x = g2 K belongs to ξˆ if and only if g2 K ∩ g1 H = ∅, which is the notion of incidence introduced by Chern in [5] and adopted by Helgason. Interchanging the roles of X and Ξ , we can define xˇ0 = K. ξ0 ⊆ Ξ,

xˇ = s(x). xˇ0 ⊆ Ξ,

where s : X → G is any section for the action (2). The notion of incidence makes clear the following duality relation x ∈ ξˆ

⇐⇒

ξ ∈ x. ˇ

ˆ 0 , we can define the Furthermore, if xˇ0 admits a relatively invariant measure m back-projection of a function fˆ : Ξ → C as R # fˆ(x) =



fˆ(s(x).ξ )dm ˆ 0 (ξ ) =:

xˇ0



fˆ(ξ )dm ˆ x (ξ ),

xˇ

provided that the integral converges, where m ˆ x is the image measure of m0 under the bijection ξ → s(x).ξ from xˇ0 onto x. ˇ The pair (X, Ξ ) is called a dual pair by Helgason if both the map ξ → ξˆ and the map x → xˇ are injective. Below we provide an alternative characterization of injectivity. We need some preliminary facts. Lemma 1 For all g ∈ G and ξ ∈ Ξ . g[ξˆ ] = g.ξ Proof For g ∈ G and ξ = g  H ∈ Ξ , by Eqs. (4) and (5) it holds that . g[ξˆ ] = gg  [ξˆ0 ] = g.ξ

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G. S. Alberti et al.

 

This concludes the proof. Lemma 2 The set  = {g ∈ G | g[ξˆ0 ] = ξˆ0 } H  ⊇ H. is a closed subgroup of G and H

 is a subgroup of G and H  ⊇ H . We prove that it is closed. Proof Clearly, H  Let (gn )n be a sequence of H converging to g and x ∈ ξˆ0 , then (gn [x])n is a sequence of ξˆ0 converging to g[x] ∈ ξˆ0 since the action is continuous and ξˆ0 is , closed. Then, g[ξˆ0 ] ⊆ ξˆ0 . The same argument applied to the sequence (gn−1 )n in H −1 −1   which converges to g , yields g [ξˆ0 ] ⊆ ξˆ0 , namely ξˆ0 ⊆ g[ξˆ0 ]. The next proposition provides an alternative characterization of the injectivity in . More precisely, the map ξ → ξˆ is injective if and only if H is chosen terms of H as the maximal subgroup fixing ξˆ0 . The reader is referred to Sect. 4 below for two examples of this aspect.  = H. Proposition 1 The map ξ → ξˆ is injective if and only if H Proof Given ξ, ξ  ∈ Ξ , the condition ξˆ = ξˆ is equivalent to σ (ξ  )−1 σ (ξ )[ξˆ0 ] = ξˆ0 , . i.e., σ (ξ  )−1 σ (ξ ) ∈ H On the other hand, since ξ = σ (ξ ).ξ0 and ξ  = σ (ξ  ).ξ0 , the condition ξ = ξ  is equivalent to σ (ξ ).ξ0 = σ (ξ  ).ξ0 , i.e., σ (ξ  )−1 σ (ξ ) ∈ H . , then since H , it follows that ξˆ = ξˆ if and only if ξ = ξ  . If H = H If H = H   is always contained in H , there exists g ∈ H \ H . Then, by Lemma 1, 0 . ξˆ0 = g[ξˆ0 ] = g.ξ However, g.ξ0 = ξ0 because g ∈ / H.

 

 is closed, we can consider the transitive space Ξ  and, since H  = G/H Since H  is a closed subgroup of H , the map , j: Ξ →Ξ

, j (gH ) = g H

, i.e., is a continuous surjection intertwining the actions of G on Ξ and Ξ j (g.ξ ) = g.j (ξ ),

g ∈ G, ξ ∈ Ξ,

∈Ξ  is still denoted by g. , we where the action of Ξ ξ . Furthermore, for all  ξ = gH define

[x0 ].  ξ = gH

Radon Transform: Dual Pairs and Irreducible Representations

11

Corollary 1 For all ξ ∈ Ξ j (ξ ) = ξˆ ,

(16)

 ξ is injective. and the map  ξ →

Proof Fix ξ = gH ∈ Ξ , then [x0 ] = g(H H )[x0 ] = g H [ξˆ0 ] = g[ξˆ0 ] = ξˆ , j (ξ ) = g H H = H , whereas the fourth equality where the second equality holds true since H  is the origin of Ξ . If ξ0 = j (ξ0 ) = H , from (16), it is due to the definition of H 

  ˆ  follows that ξ0 = ξ0 , so that H = H and the injectivity follows from Proposition 1 . with H replaced by H  

3.1 Irreducibility , then πˆ is not irreducible. In this section we show that if H is a proper subgroup of H  To prove the claim we need that H satisfies the same assumptions made on H and that there is the appropriate compatibility between the two subgroups.  has a G-relatively invariant measure d As in (A1), we first suppose that G/H ξ with the same character β of dξ . Since β satisfies β(h) =

ΔH(h) , ΔG (h)

, β(h) ∈ H

β(h) =

ΔH (h)β(h) , ΔG (h)

h ∈ H,

(17)

then ΔH (h) = ΔH(h),

h ∈ H.

(18)

/H , see [4, Equation (18) implies that there exists an invariant measure ω on H Corollary 2 Section 2, No. 6 INT VII.43]. . As in Note that ξˆ0 is a transitive space with respect to the action of H  Assumption (A2), we also assume that the measure m0 is relatively H -invariant with character  γ . Note that γ (h) = γ (h), 

h ∈ H.

, we suppose that σ is such that Finally, strengthening the analogous of (A3) for Ξ γ (σ (ξ  )−1 σ (ξ )) = 1,  for every ξ, ξ  ∈ Ξ with j (ξ ) = j (ξ  ).

(19)

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The main result of this work reads as follows.  and the Theorem 3 Under the above assumptions, if πˆ is irreducible, then H = H map ξ → ξˆ is injective. The rest of this section is devoted to the proof of this result. To prove our main result we recall the following disintegration formula. We adopt the notation of [4, Definition 1, Section 2, No. 2 INTVII.31]. Given a character β of G and a closed subgroup G0 of G, we denote by β · μG /μG0 the unique measure on the quotient space G/G0 such that for all compactly supported continuous functions f:G→C 

⎛ ⎞   ⎜ ⎟  ⎝ f (gh)dμG0 (h)⎠ d β · μG /μG0 (gG0 ).

 β(g)f (g)dμG (g) =

G

G/G0

G0

Observe first that, according to [4, Theorem 3, Section 2, No. 6 INT VII.43], the relatively invariant measures d ξ and dξ are proportional to (β · μG )/μH and (β · μG )/μH , respectively. Furthermore, note that the map ) (g, h) → gh ∈ G (G, H  onto G. The measure β·μG is right defines a continuous and proper right action of H  relatively H -invariant with character ΔH. Then by [4, Corollary 1, Section 2, No. 8 INT VII.45], there exists a positive constant C > 0 such that, for any f ∈ L1 (Ξ, dξ ) 

 f (ξ )dξ = C Ξ

⎛ ⎜ ⎝

 Ξ



⎞ ⎟ f ( σ ( ξ )h.ξ0 ) dω(hH )⎠ d ξ,

(20)

/H H

 → G is a section and the value f ( where  σ: Ξ σ ( ξ )h.ξ0 ) depends only on the left coset hH since H is the stability subgroup at ξ0 . The right-hand side is well-defined ⊂ Ξ  the map  such that if  since there is a negligible set E ξ ∈ E, /H hH → f ( H σ ( ξ )h.ξ0 ) ∈ C is integrable with respect to ω, and the almost everywhere defined function   Ξ ξ →

 /H H

f ( σ ( ξ )h.ξ0 ) ∈ C

Radon Transform: Dual Pairs and Irreducible Representations

13

is integrable with respect to d ξ . Furthermore, (20) is equivalent to 

 f (ξ )dξ = C

 Ξ

Ξ

⎛ ⎜ ⎝

⎞



⎟ f (ξ )dνξ (ξ )⎠ d ξ,

(21)

j (ξ )= ξ

where νξ is the image measure of ω under the map /H hH →  H σ ( ξ )h.ξ0 ∈ Ξ, /H onto the closed subset j −1 ( ξ ). In particular, which is a homeomorphism from H ⊂ Ξ  it holds true that the support of each νξ is j −1 ( ξ ). As a consequence, a subset E −1   is negligible with respect to dξ if and only if j (E) is negligible with respect to dξ . The next lemma shows that the Radon transform Rf (ξ ) depends only on j (ξ ). : Ξ ⊆Ξ  → C and a negligible set E  Lemma 3 For every f ∈ A there exists F  is negligible and such that j −1 (E)  ξ∈ / j −1 (E).

(j (ξ )), Rf (ξ ) = F

(22)

Furthermore, for every ξ1 , ξ2 ∈ Ξ ξˆ1 = ξˆ2

⇒

Rf (ξ1 ) = Rf (ξ2 ).

Proof Given f ∈ A , define E = {ξ ∈ Ξ : f (σ (ξ ))[·]) ∈ / L1 (ξˆ0 , m0 )}. By (7a), the set E is negligible. For any two points ξ, ξ  ∈ Ξ such that j (ξ ) = j (ξ  ),  and by assumption (19), it holds that taking into account that σ (ξ  )−1 σ (ξ ) ∈ H Rf (ξ  ) =



f (σ (ξ )σ (ξ )−1 σ (ξ  )[x]) dm0 (x)

ξˆ0

= γ (σ (ξ  )−1 σ (ξ ))

 f (σ (ξ ))[x]) dm0 (x) ξˆ0

= γ (σ (ξ  )−1 σ (ξ ))Rf (ξ ) = Rf (ξ ),  = j (E) because so that either ξ, ξ  ∈ / E or ξ, ξ  ∈ E and the claim follows with E −1  as a consequence of (21), as already E = j (j (E)). Since E is negligible, so is E observed.

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The last part immediately follows from Corollary 1.

 

As shown by the following corollary, F is the Radon transform of f associated with ), which trivially satisfies (A1)–(A4), whereas the definition of the the pair (X, Ξ Radon transform requires only (7a).  → Ξ be a measurable section such that q(ξ0 ) = ξ0 . Then Corollary 2 Let q : Ξ  → G is a measurable section. Furthermore, the pair (X, Ξ ) satisfies  σ = σ ◦q : Ξ (A3) and (7a) for all f ∈ A and, denoting the corresponding Radon transform by  for all f ∈ A , R,  (j (ξ )), Rf (ξ ) = Rf

a.e. ξ ∈ Ξ,

 ( Rf (q( ξ )) = Rf ξ ),

. a.e.  ξ ∈Ξ

and

 be the canonical projections, then p Proof Let p : G → Ξ and p : G → Ξ  = j ◦p. We readily derive p ◦  σ = j ◦ (p ◦ σ ) ◦ q = j ◦ q = Id,

 σ (j (ξ0 )) = σ (ξ0 ) = e,

 to G. so that  σ is a measurable section from Ξ  From (6), with ξ = q(ξ ) and g ∈ G we get   ι(g) = γ σ (q( ξ ))−1 gσ (g −1 .q( ξ ))     σ (g −1 .  ξ )σ (ξ  )−1 σ (ξ  ) = γ  σ ( ξ )−1 g    = γ  σ ( ξ )−1 g σ (g −1 .  ξ)  γ (σ (ξ  )−1 σ (ξ  ))   = γ  σ ( ξ )−1 g σ (g −1 .  ξ) , ξ ) and ξ  = g −1 .q( ξ ) are such that j (ξ  ) = j (ξ  ), so that the where ξ  = q(g −1 . last equality is a consequence of (19). Hence, Assumption (A3) holds true. ⊆Ξ  be the negligible set given We now prove (7a) for  σ . Take f ∈ A . Let E  then q(  Thus, for   we have by Lemma 3. Note that if  ξ∈ / E, ξ) ∈ / j −1 (E). ξ∈ /E 

|f ( σ ( ξ ))[x])|dm0 (x) =

ξˆ0



|f (σ (q( ξ ))[x])|dm0 (x) < +∞,

ξˆ0

 we have and so f ( σ ( ξ )[·]) ∈ L1 (ξˆ0 , m0 ). Similarly, for  ξ∈ /E  ( Rf ξ ) :=

  ξˆ 0

f ( σ ( ξ )[x])dm0 (x) =

 ξˆ0

f (σ (q( ξ ))[x])dm0 (x) = Rf (q( ξ )).

Radon Transform: Dual Pairs and Irreducible Representations

15

 we have j (q( Finally, for  ξ = j (ξ ) with ξ ∈ / j −1 (E) ξ )) =  ξ , and so Lemma 3 yields  (j (ξ )) = Rf (ξ ), Rf  

as desired. Lemma 4 The space (j (ξ )) for a.e. ξ ∈ Ξ for some F : Ξ  → C} L2 (Ξ, dξ )0 = {F ∈ L2 (Ξ, dξ ) | F (ξ ) = F

is a closed π-invariant ˆ subspace. Hence, if πˆ is irreducible, then L2 (Ξ, dξ )0 = L2 (Ξ, dξ ).

(23)

Proof We first observe that, given F ∈ L2 (Ξ, dξ )0 , by construction there exists : Ξ  ◦ j are equal almost everywhere. Hence, we can  → C such that F and F F  always assume that F = F ◦ j . Let (Fn ) be a sequence in L2 (Ξ, dξ )0 converging to F ∈ L2 (Ξ, dξ ). As n ◦ j where F n : Ξ  → C. Since Fn converges observed, we can assume that Fn = F to F , possibly passing to a subsequence, there exists a negligible set E such that n (j (ξ )) = F (ξ ), lim Fn (ξ ) = lim F

n→+∞

n→+∞

ξ∈ / E.

: Ξ  → C as Define F ( F ξ) =

⎧ ⎨ lim F n ( ξ)  ξ ∈ j (Ξ \ E), n→+∞

⎩0

 ξ∈ / j (Ξ \ E).

Then by construction (j (ξ )), F (ξ ) = F

ξ∈ / E.

ˆ Given It follows that L2 (Ξ, dξ )0 is closed. We now prove that it is π-invariant. g ∈ G, for all F ∈ L2 (Ξ, dξ )0 πˆ (g)F (ξ ) = β(g)−1/2 F (g −1 .ξ ) (j (g −1 .ξ )) = β(g)−1/2 F (g −1 .j (ξ )), = β(g)−1/2 F so that L2 (Ξ, dξ )0 is πˆ -invariant.

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G. S. Alberti et al.

Assume that πˆ is irreducible, then L2 (Ξ, dξ )0 is zero or the full space. Since A is not trivial, there exists a non-zero f ∈ A such that Rf ∈ L2 (Ξ, dξ )0 by (22). Furthermore, Rf = 0 since I Rf = Qf = 0 because Q is an isometry. Hence L2 (Ξ, dξ )0 is non-trivial and L2 (Ξ, dξ )0 = L2 (Ξ, dξ ).   The proof of Theorem 3 will be an immediate consequence of Proposition 1 and of the following result.  = H. Proposition 2 If πˆ is irreducible, then H Proof By Lemma 4, we have that L2 (Ξ, dξ )0 = L2 (Ξ, dξ ). Suppose by contradic = H . tion that H We first prove that ω is a finite measure. Fix f ∈ L1 (Ξ, dξ )∩L2 (Ξ, dξ ) such that  → C such that F (j (ξ )) = f (ξ ) f is positive and f = 0, then there exists F : Ξ for all ξ ∈ / E where E ⊂ Ξ is negligible. Hence, by (21) ⎛ ⎞     ⎜ ⎟ 0 < f (ξ )dξ = F (j (ξ ))dξ = C ⎝ F (j (ξ ))dνξ (ξ )⎠ d ξ Ξ

 Ξ

Ξ

 =C

j (ξ )= ξ

F ( ξ ) νξ (Ξ ) d ξ < ∞.

 Ξ

⊂ Ξ  νξ (Ξ )  such that for all  It follows that there exists a negligible subset E ξ∈ / E, is finite. By construction of νξ as image measure /H ) < +∞, νξ (Ξ ) = ω(H

 . ξ ∈Ξ

/H , the map hH → ( For every  ξ ∈Ξ σ ( ξ )h).ξ0 is a homeomorphism from H /H is not a singleton, there exist ξ1 , ξ2 ∈ j −1 ( onto j −1 ( ξ ). Thus, since H ξ ) such that ξ1 = ξ2 and, hence, two disjoint compact neighborhoods Vξ1 , Vξ2 of ξ1 and ξ2 , respectively (see Fig. 1). Since the support of νξ is j −1 ( ξ ), then νξ (Vξ1 ) > 0,

νξ (Vξ2 ) > 0.

(24)

 be a compact set of positive measure, Let now K ⊂ Ξ Z = {ξ ∈ Ξ : j (ξ ) ∈ K, ξ ∈ Vj1(ξ ) }, and f be the characteristic function of Z. By applying (21) we obtain     /H ) d 0 < |f (ξ )|2 dξ = f (ξ )dξ = C νξ (Vξ1 )d ξ ≤ Cω(H ξ < +∞, Ξ

Ξ

K

K

/H ) < +∞ and K is compact. We can apply (21) since which is finite since ω(H f is positive [4, item c), Corollary 1, Section 2, No. 8 INT VII.45]. Hence f ∈

Radon Transform: Dual Pairs and Irreducible Representations

17

Fig. 1 The setup considered in the proof of Proposition 2

 → C such that F (j (ξ )) = f (ξ ) for all L2 (Ξ, dξ ) and, as above, there exists F : Ξ   ξ ∈ / E where E ⊂ Ξ is negligible. Since E  is negligible, by (21) applied to the  ⊂ Ξ  such that characteristic function of E  there exists a negligible subset E   ξ ) = 0, νξ E  ∩ j −1 (

 .  ξ∈ /E

(25)

 . By (24) and (25), for i = 1, 2 there exists ξ i ∈ Choose an arbitrary  ξ ∈ K\E i  i   . Thus Vξ \ E such that j (ξ ) = E 

1 F ( ξ ) = F (j (ξ i )) = f (ξ i ) = 0

if i = 1, if i = 2,

which is absurd.

 

4 3D-Signals: Radon and Ray Transforms 4.1 The Radon Transform on R3 4.1.1

Groups and Spaces

The Radon transform on R3 of a signal f is defined as the integral of f over the set of planes in R3 . We show that this is an example of our construction.

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G. S. Alberti et al.

The input space is X = R3 and the group is SIM(3), the semi-direct product = {aR ∈ GL(3, R) : R ∈ SO(3), a ∈ R+ }. Under the identification K  SO(3) × R+ , we write (b, R, a) for the elements in SIM(3) and the group law becomes R3 K, with K

(b, R, a)(b , R  , a  ) = (b + aRb , RR  , aa  ). A left Haar measure of SIM(3) is given by dμ(b, R, a) = a −4 dbdRda,

(26)

where db and da are the Lebesgue measures on R3 and R+ , respectively, and dR is a Haar measure of SO(3). The group SIM(3) acts on R3 by the canonical action (b, R, a)[x] = b + aRx,

(b, R, a) ∈ SIM(3), x ∈ R3 .

The isotropy at the origin x0 = 0 is the subgroup {(0, k) : k ∈ K} which we identify with K, so that X  SIM(3)/K. Furthermore, the Lebesgue measure dx on R3 is a relatively SIM(3)-invariant measure with positive character α(b, R, a) = a 3 . It remains to choose the closed subgroup H of SIM(3) in such a way that {

ξ } is the set of planes in R3 . We consider H = (R2 × {0})  (O(2) × R+ ), where O(2) denotes the subgroup of rotations leaving the plane z = 0 invariant, i.e., it consists of the matrices of the form   R ± Rθ 0 R= , 0 ±1 0 ]. By (3), the root manifold where Rθ ∈ SO(2), R+ is the identity and R− = [ 10 −1 is the xy-plane

ξˆ0 = H [x0 ] = {x ∈ R3 : x · e3 = 0} and it is easy to verify that m0 = dxdy is a relatively H -invariant measure on ξˆ0 with character γ (b, R, a) = a 2 . Furthermore, for each ξ = (b, R, a)H ∈ Ξ = SIM(3)/H , by (4) we compute ξˆ = (b, R, a)[ξˆ0 ] = {x ∈ R3 : Re3 · x = Re3 · b}, which is the plane perpendicular to the vector Re3 and passing through b. It is worth observing that H is the maximal closed subgroup of SIM(3) which satisfies h[ξˆ0 ] = ξˆ0 for every h ∈ H and then, by Proposition 1, the map ξ → ξˆ is injective.  and, Had we chosen H  = (R2 × {0})  (SO(2) × R+ ), we would have had H   H indeed, the map ξ  → ξ  would not have been injective, since every plane would have been labeled by two different ξ  .

Radon Transform: Dual Pairs and Irreducible Representations

19

We identify the coset space Ξ = SIM(3)/H with the set [0, π )2∗ × R, where   [0, π )2∗ = [0, π ) × (0, π ) ∪ {(0, 0)}, and we write n(θ, ϕ) = t (sin ϕ cos θ, sin ϕ sin θ, cos ϕ) for every (θ, ϕ) ∈ [0, π )2∗ . The group SIM(3) acts on [0, π )2∗ × R by the transitive action (b, R, a).(θ, ϕ, t) = (θR , ϕR , R n(θ, ϕ) · n(θR , ϕR )(at + R n(θ, ϕ) · b)), where (θR , ϕR ) ∈ [0, π )2∗ is such that R n(θ, ϕ) = ±n(θR , ϕR ). Since the stability subgroup at (0, 0, 0) is H , then SIM(3)/H  [0, π )2∗ × R under the canonical isomorphism (b, R, a)H → (b, R, a).(0, 0, 0). We endow Ξ with the measure dξ = sin ϕ dθ dϕdt, where dθ , dϕ, and dt are the Lebesgue measure on [0, π ) and R, respectively. It is easy to verify that dξ is a relatively SIM(3)-invariant measure on Ξ with positive character β(b, R, a) = a.

4.1.2

The Representations

The group SIM(3) acts on L2 (R3 ) by means of the unitary representation π defined by 3

π(b, R, a)f (x) = a − 2 f (a −1 R −1 (x − b)).

(27)

The dual action R3 × K (η, k) → t kη has a single open orbit O = R3 for η = e3 of full measure and the stabilizer Ke3  SO(2) × {1} is compact. Then, the representation π is irreducible and square-integrable, see [2]. Furthermore, the quasi-regular representation πˆ of SIM(3) acting on L2 (Ξ, dξ ) as   1 t −n(θ, ϕ) · b π(b, ˆ R, a)F (θ, ϕ, t)=a − 2 F θR −1 , ϕR −1 , R −1 n(θ, ϕ) · n(θR −1 , ϕR −1 ) a

is irreducible, too. As a consequence, Theorem 3 guarantees that the map ξ → ξˆ is injective. Let us consider again the situation with the choice H  = (R2 × {0})  (SO(2) × R+ ). In this case, Ξ  = SIM(3)/H  may be identified with S 2 × R, and we have already observed that the map ξ  = (n, t) → ξˆn,t = {x ∈ R3 : n · x = t} is not injective, since (n1 , t1 ) and (n2 , t2 ) identify the same plane if

20

G. S. Alberti et al.

(n1 , t1 ) = ±(n2 , t2 ).

(28)

In the notation of Sect. 3, this corresponds to j (n1 , t1 ) = j (n2 , t2 ). According to Theorem 3, this implies that the corresponding quasi-regular representation π  cannot be irreducible. Let us verify this explicitly, in order to visualize the link between the irreducibility of π  and the injectivity of ξ  → ξ  in this example. By arguing as above, it is easy to prove that   1 t −n·b . π  (b, R, a)F (n, t) = a − 2 F R −1 n, a Thus, using the notation of Lemma 4, the set L2 (Ξ  , dξ  )0 = {F ∈ L2 (Ξ  , dξ  ) : F (n1 , t1 ) = F (n2 , t2 ) if (28) holds} is a closed π  -invariant proper subspace of L2 (Ξ  , dξ  ). Hence, π  is not irreducible.

4.1.3

The Radon Transform

In order to define the Radon transform we need to endow each ξˆ with a suitable measure. Since the measure m0 is H -relatively invariant, the choice of the representative of ξ is crucial. We fix the Borel section σ : Ξ → G,

σ (θ, ϕ, t) = (tn(θ, ϕ), Rθ,ϕ , 1),

with Rθ,ϕ ∈ SO(3) such that Rθ,ϕ e3 = n(θ, ϕ). We observe that, since γ extends to a positive character of G, Assumption (A4) is implied by the stronger condition γ (σ (ξ )) = 1 for every ξ ∈ Ξ . Then, we compute the Radon transform by (9) obtaining  Rf (θ, ϕ, t) =

  f tn(θ, ϕ) + Rθ,ϕ (x, y, 0) dxdy,

(29)

R2

which is the integral of f on the plane of equation n(θ, ϕ) · x = t. As a consequence of Fubini theorem, Eq. (29) makes sense, for instance, if f ∈ L1 (R3 ). We recall a crucial result in Radon transform theory in its standard version, known as Fourier slice theorem. We denote by I the identity operator. Proposition 3 For every f ∈ L1 (R3 ) (I ⊗ F )Rf (θ, ϕ, τ ) = F f (τ n(θ, ϕ)), for all (θ, ϕ, τ ) ∈ [0, π )2∗ × R.

(30)

Radon Transform: Dual Pairs and Irreducible Representations

21

Here the Fourier transform on the right-hand side is in R3 , whereas the operator F on the left-hand side is one-dimensional and acts on the variable t. We repeat this slight abuse of notation in other formulas below. We show that Assumption (A6) holds true. Let S 2 be the sphere in R3 and denote by S (R3 ) and S (S 2 × R) the Schwartz spaces of rapidly decreasing functions on R3 and on S 2 × R, respectively, and by S  (R3 ) and S  (S 2 × R) the corresponding spaces of tempered distributions; see [11, Chapter 1.2] for the definition on S 2 × R. We extend the Radon transform R as an even function on S 2 and we denote it by Re , i.e., Re f (u, t) = Rf (θu , ϕu , u · n(θu , ϕu )t), where (θu , ϕu ) ∈ [0, π )2∗ is such that n(θu , ϕu ) = ±u. We recall that, since Re is a continuous map from S (R3 ) into S (S 2 × R) (see [9]), given F ∈ S  (S 2 ×R), the tempered distribution Re# F : S (R3 ) → C given by Re# F, f = F, Re f is well-defined. If F ∈ S (S 2 ×R), by Theorem 1.4 in [15, Chapter 2], the tempered distribution F Re# F is represented by the function F Re# F (v) = |v|−2 [(I ⊗ F )F (v/|v|, |v|) + (I ⊗ F )F (−v/|v|, −|v|)].

(31)

By Eq. (31), Re# F is in L2 (R3 ) provided that  t m F (u, t)dt = 0,

m ∈ N.

(32)

R

We fix a non-zero F ∈ S (S 2 × R) which satisfies (32) and the symmetry condition F (u, t) = F (−u, −t) and we denote its restriction to [0, π )2∗ × R by F0 , that is F0 (θ, ϕ, t) = F (n(θ, ϕ), t), for every (θ, ϕ, t) ∈ [0, π )2∗ × R. Then, there exists a positive constant C such that |F0 , Rf L2 ([0,π )2∗ ×R) | =

1 |F, Re f L2 (S 2 ×R) | = |Re# F, f | ≤ C f , 2

for any f ∈ S (R3 ). Therefore, if we take f0 ∈ S (R3 ) and define the vector subspace A = span{π(g)f0 : g ∈ G} ⊆ S (R3 ), then the domain of the adjoint of the restriction of R to A is non-trivial since F0 ∈ dom(R ∗ ) and Assumption (A6) holds true.

22

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G. S. Alberti et al.

The Unitarization Theorem

By Theorem 1, the Radon transform R : A → L2 (Ξ, dξ ) admits a unique closure R which satisfies Rπ(b, R, a) = χ (b, R, a)−1 πˆ (b, R, a)R,

(b, R, a) ∈ G,

(33)

where χ (b, R, a) = a since α(b, R, a) = a 3 , β(b, R, a) = a and γ (b, R, a) = a 2 . Furthermore, there exists a unique positive self-adjoint operator I : dom(I ) ⊇ Im R → L2 (Ξ, dξ ), semi-invariant with weight χ (b, R, a)−1 = a −1 with the property that the composite operator I R extends to a unitary operator Q : L2 (X, dx) → L2 (Ξ, dξ ) intertwining π and πˆ , namely πˆ (g)Qπ(g)−1 = Q,

g ∈ G.

(34)

We can provide an explicit formula for I . Consider the subspace  D={f ∈ L2 ([0, π )2∗ ×R) : |τ |2 |(I ⊗F )f (θ, ϕ, τ )|2 sin ϕ dθ dϕdτ < +∞} [0,π )2∗ ×R

and define the operator J : D → L2 ([0, π )2∗ × R) by (I ⊗ F )J f (θ, ϕ, τ ) = |τ |(I ⊗ F )f (θ, ϕ, τ ),

(35)

a Fourier multiplier with respect to the variable t. A direct calculation shows that J is a densely defined positive self-adjoint injective operator and is semi-invariant with weight ζ (g) = χ (g)−1 = a −1 . By [6, Theorem 1], there exists c > 0 such that I = cJ and we now show that c = 1. Take a non-zero function f ∈ A . Then, by Plancherel theorem and Proposition 3 we have that

f 2 = I Rf 2L2 ([0,π )2 ×R) = c2 (I ⊗ F )J Rf 2L2 ([0,π )2 ×R) ∗ ∗  = c2 |(I ⊗ F )Rf (θ, ϕ, τ )|2 |τ |2 sin ϕ dθ dϕdτ [0,π )2∗ ×R



= c2

|F f (τ n(θ, ϕ))|2 |τ |2 sin ϕ dθ dϕdτ [0,π )2∗ ×R

= c2 f 2 . Thus, we obtain c = 1.

Radon Transform: Dual Pairs and Irreducible Representations

4.1.5

23

The Inversion Formula

By Theorem 2, for any f ∈ A we have the reconstruction formula  9 f = a − 2 Rf, πˆ (b, R, a)Ψ L2 (Ξ,dξ ) ψ(a −1 R −1 (x − b))dbdRda, SIM(3)

where the integral is weakly convergent and where we used that χ (b, R, a) = a, the expression of the Haar measure of SIM(3) given in (26) and the expression of π given in (27).

4.2 The X-ray Transform The X-ray transform in the Euclidean 3-space maps a function on R3 into the set of integrals over the lines and the X-ray reconstruction problem consists in reconstructing a signal f by means of its line integrals.

4.2.1

Groups and Spaces

Take the same group G = SIM(3) as in Sect. 4.1, namely G = R3  K, with K = {aR ∈ GL(3, R) : R ∈ SO(3), a ∈ R+ }. Firstly, we choose X = R3 and, for what concerns this space, we keep the notation as in Sect. 4.1. Then, we consider the space Ξ = G/H , where H = ({(0, 0)} × R)  (O(2) × R+ ). By (3), the root manifold is then ξˆ0 = {te3 : t ∈ R} and it is easy to verify that m0 = dt is a relatively H -invariant measure on ξˆ0 with character γ (b, R, a) = a. Furthermore, for each ξ = (b, R, a)H ∈ Ξ , by (4) we compute ξˆ = (b, R, a)[ξˆ0 ] = {tRe3 + b : t ∈ R}, which is the line parallel to the vector Re3 and passing through the point b. It is worth observing that H is the maximal closed subgroup of SIM(3) which satisfies h[ξˆ0 ] = ξˆ0 for every h ∈ H and then, by Proposition 1, the map ξ → ξˆ is injective. The coset space Ξ = SIM(3)/H can be identified with the set T = {(θ, ϕ, t) : (θ, ϕ) ∈ [0, π )2∗ , t ∈ (θ, ϕ)⊥ }, where (θ, ϕ)⊥ denotes the plane passing through the origin and perpendicular to the vector n(θ, ϕ), i.e., the plane of equation n(θ, ϕ)·x = 0. The group SIM(3) acts on T by the action (b, R, a).(θ, ϕ, t) = (θR , ϕR , t + aRb − (n(θR , ϕR ) · (t + aRb))n(θR , ϕR )),

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where we recall that (θR , ϕR ) ∈ [0, π )2∗ is such that R n(θ, ϕ) = ±n(θR , ϕR ). Since the stability subgroup at (0, 0, 0) is H , then SIM(3)/H  T under the canonical isomorphism (b, R, a)H → (b, R, a).(0, 0, 0). We endow Ξ with the measure dξ = sin ϕ dθ dϕdt, with dθ , dϕ, and dt being the Lebesgue measure on [0, π ) and R3 , respectively. It is easy to verify that dξ is a relatively SIM(3)-invariant measure on Ξ with positive character β(b, R, a) = a 3 .

4.2.2

The Representations

We recall that the group SIM(3) acts on L2 (R3 ) by means of the unitary irreducible representation π defined by 3

π(b, R, a)f (x) = a − 2 f (a −1 R −1 (x − b)). Furthermore, the quasi-regular representation πˆ of SIM(3) acting on L2 (Ξ, dξ ) as πˆ (b, R, a)F (θ, ϕ, t)=   3 a − 2 F θR −1 , ϕR −1 , a −1 R −1 (t − b)−(n(θR − 1 , ϕR − 1 ) · a −1 R −1 (t − b))n(θR − 1 , ϕR − 1 )

is irreducible, too.

4.2.3

The Radon Transform

We fix the Borel section σ : Ξ → G,

σ (θ, ϕ, t) = (t, Rθ,ϕ , 1),

with Rθ,ϕ ∈ SO(3) such that Rθ,ϕ e3 = n(θ, ϕ). We observe that, since γ extends to a positive character of G, Assumption (A4) is implied by the stronger condition γ (σ (ξ )) = 1 for every ξ ∈ Ξ . Then, we compute by (9) the Radon transform between the SIM(3)-transitive spaces X and Ξ obtaining  Rf (θ, ϕ, t) =

f (tn(θ, ϕ) + t)dt,

(36)

R

which is the integral of f over the line parallel to the vector n(θ, ϕ) and passing through the point t ∈ R3 . Let us now determine a suitable π -invariant subspace A of L2 (R3 ) as in (A7). In order to do that, it is useful to derive a Fourier slice theorem for R.

Radon Transform: Dual Pairs and Irreducible Representations

25

For any f ∈ S (R3 ), by Theorem 1.1 in [15, Chapter 2], we have (I ⊗ F )Rf (θ, ϕ, v) = F f (v),

v ∈ (θ, ϕ)⊥ .

(37)

As a consequence, by Plancherel theorem and formula (2.8) in [15, Chapter 7], we obtain  π π 

Rf 2L2 (Ξ ) = |(I ⊗ F )Rf (θ, ϕ, v)|2 sin ϕ dvdθ dϕ 0



0



π

= 0

 = R3

(θ,ϕ)⊥



π

|F f (v)|2 sin ϕ dvdθ dϕ

0

(θ,ϕ)⊥

|F f (v)|2 dv. |v|

By using spherical coordinates, we obtain 

Rf 2L2 (Ξ ) =

π



0



0 π

≤



0

 + 0

π

0

|F f (τ n(θ, ϕ))|2 |τ | sin ϕ dτ dθ dϕ R

π

0

π



π

 |F f (τ n(θ, ϕ))|2 sin ϕ dτ dθ dϕ

|τ |≤1



|τ ||F f (τ n(θ, ϕ))|2 sin ϕ dτ dθ dϕ |τ |>1

≤ 4π f 21 + f 22 < +∞, which proves that Rf ∈ L2 (Ξ ) for any f ∈ S (R3 ) so that we can set A = S (R3 ). Next, we show that R, regarded as an operator from A to L2 (Ξ ), is closable. By [18, Theorem VIII.1], this is equivalent to proving that the adjoint of Rf : A → L2 (Ξ ) is densely defined. Suppose that (fn )n ⊆ A is a sequence such that fn → f in L2 (R3 ) and Rfn → g in L2 (Ξ ). Since I ⊗ F is unitary from L2 (Ξ ) onto L2 (Ξ ), we have that (I ⊗ F )Rfn → (I ⊗ F )g in L2 (Ξ ). Since fn ∈ A , by (37), for every (θ, ϕ) ∈ [0, π )2∗ (I ⊗ F )Rfn (θ, ϕ, v) = F fn (v),

v ∈ (θ, ϕ)⊥ .

Hence, passing to a subsequence if necessary, F fn (v) → (I ⊗ F )g(θ, ϕ, v)

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for almost every (θ, ϕ) ∈ [0, π )2∗ and v ∈ (θ, ϕ)⊥ . Therefore, for almost every (θ, ϕ) ∈ [0, π )2∗ and v ∈ (θ, ϕ)⊥ (I ⊗ F )g(θ, ϕ, v) = lim F fn (v) = F f (v), n→+∞

where the last equality holds true using a subsequence if necessary. Therefore, if (hn )n ∈ A is another sequence such that hn → f in L2 (R3 ) and Rhn → h in L2 (Ξ ), then for almost every (θ, ϕ) ∈ [0, π )2∗ and v ∈ (θ, ϕ)⊥ (I ⊗ F )h(θ, ϕ, v) = F f (v). Therefore, (I ⊗ F )g(θ, ϕ, v) = (I ⊗ F )h(θ, ϕ, v) for almost every (θ, ϕ) ∈ [0, π )2∗ and v ∈ (θ, ϕ)⊥ . Then lim Rfn = lim Rhn , n→+∞

n→+∞

and R is closable. We denote its closure by R.

4.2.4

The Unitarization Theorem

By Theorem 1, the Radon transform R : A → L2 (Ξ, dξ ) admits a unique closure R which satisfies R π(b, R, a) = χ (b, R, a)−1 πˆ (b, R, a) R,

(b, R, a) ∈ G,

(38)

where χ (b, R, a) = a since α(b, R, a) = a 3 , β(b, R, a) = a and γ (b, R, a) = a 2 . Furthermore, there exists a unique positive self-adjoint operator I : dom(I ) ⊇ Im R → L2 (Ξ, dξ ), semi-invariant with weight χ (b, R, a)−1 = a −1 with the property that the composite operator I R extends to a unitary operator Q : L2 (X, dx) → L2 (Ξ, dξ ) intertwining π and πˆ , namely πˆ (g) Q π(g)−1 = Q,

g ∈ G.

(39)

We can provide an explicit formula for I . Consider the subspace  D = {f ∈ L (Ξ ) :

|τ ||(I ⊗ F )f (θ, ϕ, τ )|2 sin ϕ dθ dϕdτ < +∞}

2

[0,π )2∗ ×(θ,ϕ)⊥

Radon Transform: Dual Pairs and Irreducible Representations

27

and define the operator J : D → L2 (Ξ ) by 1

(I ⊗ F )J f (θ, ϕ, τ ) = |τ | 2 (I ⊗ F )f (θ, ϕ, τ ),

(40)

a Fourier multiplier with respect to the variable t. A direct calculation shows that J is a densely defined positive self-adjoint injective operator and is semi-invariant 1 with weight ζ (g) = χ (g)−1 = a − 2 . By [6, Theorem 1], there exists c > 0 such that I = cJ and we now show that c = 1. Consider a non-zero function f ∈ A . Then, by Plancherel theorem, Eq. (37) and formula (2.8) in [15, Chapter 7], we obtain

f 2 = I Rf 2L2 (Ξ ) = c2 (I ⊗ F )J Rf 2L2 (Ξ )  = c2 |(I ⊗ F )Rf (θ, ϕ, τ )|2 |τ | sin ϕ dθ dϕdτ [0,π )2∗ ×(θ,ϕ)⊥



= c2

|F f (τ )|2 |τ | sin ϕ dθ dϕdτ [0,π )2∗ ×(θ,ϕ)⊥

= c2 f 2 . Thus, c = 1 and this concludes the proof.

4.2.5

The Inversion Formula

By Theorem 2, for any f ∈ S (R3 ), taking into account Eqs. (26) and (27) and that 1 χ (b, R, a) = a 2 , the reconstruction formula (14) reads  a −5 Rf, πˆ (b, R, a)Ψ L2 (Ξ,dξ ) ψ(a −1 R −1 (x − b))dbdRda, f = SIM(3)

where the integral is weakly convergent. Acknowledgments G.S. Alberti, F. De Mari, and E. De Vito are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and together with F. Bartolucci are part of the Machine Learning Genoa Center (MaLGa).

References 1. G. S. Alberti, F. Bartolucci, F. De Mari, E. De Vito. Unitarization and inversion formulae for the Radon transform between dual pairs. SIAM J. Math. Anal. 51 (2019), no. 2, 4356–4381. 2. J.P. Antoine and R. Murenzi. Two-dimensional directional wavelets and the scale-angle representation. Signal Process. 52 (1996), no. 3, 259–281.

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3. C. Berenstein and D. Walnut. Local inversion of the Radon transform in even dimensions using wavelets. 75 years of Radon transform, Vienna, 1992, Conf. Proc. Lecture Notes Math. Phys., IV, Int. Press, Cambridge, MA (1994), 45–69. 4. N. Bourbaki. Integration II . Springer-Verlag, Berlin (2004). 5. S. Chern. On integral geometry in Klein spaces. Ann. of Math. (2) 43 (1942), 178–189. 6. M. Duflo and C. C. Moore. On the regular representation of a nonunimodular locally compact group. J. Funct. Anal. 21 (1976), no. 2, 209–243. 7. H. Führ. Abstract Harmonic Analysis of Continuous Wavelet Transforms. Springer, Berlin (2005). 8. I. M. Gel’fand. Integral geometry and its relation to the theory of representations. Russ. Math. Surv. 15 (1960), no. 2, 143–151. 9. S. Helgason. The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113 (1965), 153–180. 10. S. Helgason. Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs, American Mathematical Society 39, Providence, RI, 2nd ed. (2008). 11. S. Helgason. The Radon Transform. Progress in Mathematics 5, Birkhäuser Boston, Inc., Boston, MA, 2nd ed. (1999). 12. S. Helgason. Some personal remarks on the Radon transform. Geometric analysis and integral geometry. Contemp. Math. 598, Amer. Math. Soc., Providence, RI (2013), 3–19. 13. M. Holschneider. Inverse Radon transforms through inverse wavelet transforms. Inverse Probl. 7 (1991), 853–861. 14. W. R. Madych. Tomography, approximate reconstruction, and continuous wavelet transforms. Appl. Comput. Harmon. Anal. 7 (1999), 54–100. 15. F. Natterer. The Mathematics of Computerized Tomography. SIAM (2001). 16. T. Olson and J. DeStefano. Wavelet localization of the Radon transform. IEEE Trans. Sign. Proc. 42 (1994), 2055–2067. 17. J. Radon. Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Ver. Sächs. Akad. Wiss. Leipzig, Math Nat. Kl. 69 (1917), 262–277. 18. M. Reed and B. Simon. Methods of Modern Mathematical Physics. I: Functional Analysis. Academic Press, New York, 2nd ed. (1980). 19. H. Reiter. Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (1968). 20. B. Rubin. The Calderón reproducing formula, windowed X-ray transforms, and Radon transforms in Lp -spaces. J. Fourier Anal. Appl. 4 (1998), 175–197. 21. D. C. Solmon. The X-Ray transform. J. Math. Anal. Appl. 56 (1976), no. 1, 61–83. 22. V. S. Varadarajan. Geometry of Quantum Theory. Springer-Verlag, New York, 2nd ed. (1985). 23. D. Walnut. Local inversion of the Radon transform in the plane using wavelets. Proc. SPIE 2034 (1993), 84–90. 24. C. E. Yarman and B. Yazici. Euclidean motion group representations and the singular value decomposition of the Radon transform. Integral Transforms Spec. Funct. 18 (2007), 59–76.

Data Approximation with Time-Frequency Invariant Systems Davide Barbieri, Carlos Cabrelli, Eugenio Hernández, and Ursula Molter

Abstract In this paper we prove the existence of a time-frequency space that best approximates a given finite set of data. Here best approximation is in the least square sense, among all time-frequency spaces with no more than a prescribed number of generators. We provide a formula to construct the generators from the data and give the exact error of approximation. The setting is in the space of square integrable functions defined on a second countable LCA group and we use the Zak transform as the main tool. Keywords Time-frequency space · Eckart–Young Theorem · LCA groups · Zak transform

1 Introduction and Main Result Time-frequency systems, also called Gabor or Weyl-Heisenberg systems in the literature, are used extensively in the theory of communication, to analyze continuous signals, and to process digital data such as sampled audio or images. Time-frequency spaces try to represent features of both a function and its frequencies by decomposing the signal into time-frequency atoms given by modulations and translations of a finite number of functions [9]. If one looks at a musical score, on the horizontal axis the composer represents the time, and on the vertical axis the “frequency” given by the amplitude of the signal at that instant. Finding sparse representations (i.e., spaces generated by a small set of functions) will be useful, for example, in classification tasks.

D. Barbieri · E. Hernández Universidad Autónoma de Madrid, Madrid, Spain e-mail: [email protected]; [email protected] C. Cabrelli · U. Molter () Departamento de Matemática, Universidad de Buenos Aires, and Instituto de Matemática “Luis Santaló” (IMAS-CONICET-UBA), Buenos Aires, Argentina e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_2

29

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In numerical applications to time-dependent phenomena, one often encounters uniformly sampled signals of finite length, i.e., vectors of d elements, such as audio signals with a constant sampling frequency. In this case the most direct approach is to consider Fourier analysis on the cyclic group Zd . To include a large variety of situations, our setting will be that of a locally compact abelian (LCA) group. The general construction developed in this paper will be specialized to the cyclic group Zd in Example 2. In this paper G = (G, +) will be a second countable LCA group, that is, an abelian group endowed with a locally compact and second countable Hausdorff topology for which (x, y) → x − y is continuous from G × G into G. We denote

the dual group of G, formed by the characters of G: an element α ∈ G

is a by G continuous homomorphism from G into T = {z ∈ C : |z| = 1}. The action of α on

is x ∈ G will be denoted by (x, α) := α(x), to reflect the fact that the dual of G

isomorphic to G, and therefore x can also act on α. For α1 , α2 ∈ G the group law is denoted by α1 · α2 , so that (x, α1 · α2 ) = (x, α1 )(x, α2 ). A uniform lattice, L ⊂ G, is a subgroup of G whose relative topology is the discrete one and for which G/L is compact in the quotient topology. The annihilator  [11, Theorem

: (, α) = 1, ∀ ∈ L}. Since L⊥ ≈ (G/L) of L is L⊥ = {α ∈ G 2.1.2] and G/L is compact, L⊥ is discrete [11, Theorem 1.2.5]. In particular, since

is also second countable, so both discrete groups L and G is second countable, G L⊥ are countable. Let L be a uniform lattice in the LCA group G and B ⊂ L⊥ be a uniform

For f ∈ L2 (G),  ∈ L, and β ∈ B let T f (x) = lattice in the dual group G. f (x − ), x ∈ G, be the translation operator, and Mβ f (x) = (x, β)f (x), x ∈ G, be the modulation operator. The collection {T Mβ f :  ∈ L, β ∈ B} is the time-frequency system generated by f ∈ L2 (G). Since B ⊂ L⊥ , we have T Mβ f = Mβ T f for all f ∈ L2 (G),  ∈ L, and β ∈ B. Thus Π (, β) := T Mβ

(1)

is a unitary representation of the abelian group Γ := L×B, with operation (1 , β1 )· (2 , β2 ) = (1 + 2 , β1 · β2 ), in L2 (G). A closed subspace V of L2 (G) is said to be Γ -invariant (or time-frequency invariant) if for every f ∈ V , Π (, β)f ∈ V for every (, β) ∈ Γ. All Γ -invariant subspaces V of L2 (G) are of the form L2 (G)

V = SΓ (A ) := span{T Mβ ϕ : ϕ ∈ A , (, β) ∈ Γ }

for some countable collection A of elements of L2 (G). If A is a finite collection we say that V = SΓ (A ) has finite length, and A is a set of generators of V . We

Time-Frequency Data Approximation

31

call the length of V , denoted length(V ), the minimum positive integer n such that V has a set of generators with n elements. We now state our approximation problem. Let F = {f1 , f2 , . . . , fm } ⊂ L2 (G) be a set of functional data. Given a closed subspace V of L2 (G) define E (F ; V ) :=

m  j =1

fj − PV fj 2L2 (G)

(2)

as the error of approximation of F by V , where PV denotes the orthogonal projection of L2 (G) onto V . Is it possible to find a Γ -invariant space of length at most n < m that best approximates our functions, in the sense that E (F ; SΓ {ψ1 , . . . , ψn }) ≤ E (F ; V ) for all Γ -invariant subspaces V of L2 (G) with length(V ) ≤ n? This question is relevant in applications. For example, if {f1 , . . . , fm } are audio signals, the best Γ -invariant space provides a time-frequency optimal model to represent these signals. The answer to this question is affirmative, and is given by the main theorem of this work. Theorem 1 Let G be a second countable LCA group, L and B be uniform lattices

respectively, with B ⊂ L⊥ . For each set of functional data F = in G and G, {f1 , f2 , . . . , fm } ⊂ L2 (G) and each n ∈ N, n < m, there exists {ψ1 , . . . , ψn } ⊂ L2 (G) such that E (F ; SΓ {ψ1 , . . . , ψn }) ≤ E (F ; V ) for all Γ -invariant subspaces V of L2 (G) with length(V ) ≤ n. Remark Observe that, in the previous statement, some of the generators {ψ1 , . . . , ψn } may be zero. In this case, the length of SΓ {ψ1 , . . . , ψn } would be strictly smaller than n. The proof of Theorem 1 will follow the ideas originally developed in [1] for approximating data in L2 (Rd ) by shift-invariant subspaces of finite length, and which have also been used in [2, 6]. We reduce the problem of finding the collection {ψ1 , . . . , ψn }, whose existence is asserted in Theorem 1, to solve infinitely many approximation problems for data in a particular Hilbert space of sequences. This is accomplished with the help of an isometric isomorphism HΓ that intertwines the unitary representation Π with the characters of Γ . This isometry HΓ generalizes the fiberization map of [4] used in [1], and has the properties of a Helson map as defined in [3, Definition 7]. The definition and properties of HΓ are given in Sect. 2.

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The reduced problems are then solved by using Eckart–Young Theorem as stated and proved in [1] (Theorem 4.1). The solutions of all of these reduced problems are patched together to finally obtain the proof of Theorem 1 in Sect. 3.

2 An Isometric Isomorphism Let G be a second countable LCA group, L a uniform lattice in G, and B ⊂ L⊥ a

(see definitions in Sect. 1). With Γ = L × B, each Γ -invariant uniform lattice in G 2 subspace V of L (G) is of the form L2 (G)

V = SΓ (A ) := span{T Mβ ϕ : ϕ ∈ A , (, β) ∈ Γ } for some countable set A ⊂ L2 (G). Therefore V = SL ({Mβ ϕ : ϕ ∈ A , β ∈ B})

is also an L-invariant subspace, that is, T f ∈ V for all  ∈ L whenever f ∈ V . The theory of shift-invariant spaces on LCA groups, as developed in [7], can be applied to this situation.

be a measurable cross-section of G/L

⊥ . The set ΩL⊥ is in one to Let ΩL⊥ ⊂ G ⊥

one correspondence with the elements of G/L , and {ΩL⊥ + λ : λ ∈ L⊥ } is a tiling

of G.  Let f (ω) := G f (x)(x, w)dx denote the unitary Fourier transform of f ∈ L2 (G) ∩ L1 (G) and extended to L2 (G) by density. By Proposition 3.3 in [7] the mapping T : L2 (G) → L2 (ΩL⊥ , 2 (L⊥ )) given by T f (ω) = {f (ω + λ)}λ∈L⊥ , f ∈ L2 (G)

(3)

is an isometric isomorphism. Moreover, since V ⊂ L2 (G) is an L-invariant space, it has an associated measurable range function J : ΩL⊥ −→ {closed subspaces of 2 (L⊥ )} such that (See Theorem 3.10 in [7]) 2 (L⊥ )

J (ω) = span {T (Mβ ϕ)(ω) : β ∈ B, ϕ ∈ A }

, a.e. ω ∈ ΩL⊥ .

(4)

Using the definition of T given in (3), for each β ∈ B and each ϕ ∈ L2 (G) we have  ϕ (ω +λ−β)}λ∈L⊥ = tβ (T ϕ(ω)), T (Mβ ϕ)(ω) = {M β ϕ (ω +λ)}λ∈L⊥ = {

(5)

Time-Frequency Data Approximation

33

where tβ : 2 (L⊥ ) −→ 2 (L⊥ ) is the translation of sequences in 2 (L⊥ ) by elements of β ∈ B, that is, tβ ({a(λ)}λ∈L⊥ ) = {a(λ − β)}λ∈L⊥ . Therefore, T intertwines the modulations {Mβ }β∈B with the translations by B on 2 (L⊥ ). By Eqs. (4) and (5), for a. e. ω ∈ ΩL⊥ , J (ω) = span {tβ (T ϕ(ω)) : β ∈ B, ϕ ∈ A }

2 (L⊥ )

.

Therefore, J (ω) is a B-invariant subspace of L2 (L⊥ ). We can apply the theory of shift-invariant spaces as developed in [7] to the discrete LCA group L⊥ and its uniform lattice B.  ⊥ ⊂ G, that is, Let B ⊥ be the annihilator of B in the compact group L  ⊥ : (b, β) = 1, ∀β ∈ B}. B ⊥ = {b ∈ L

(6)

Observe that B ⊥ is finite, because it is a discrete subgroup of a compact group.   ⊥ be a measurable cross-section of L ⊥ /B ⊥ . The set Ω ⊥ is in Let ΩB ⊥ ⊂ L B ⊥  one to one correspondence with the elements of L⊥ /B and {ΩB ⊥ + b : b ∈ B ⊥ }  ⊥. is a tiling of L  ⊥ =

Example 1 Let G = R, L = Z, and B = nZ ⊂ L⊥ = Z ⊂

R. Since L Z≈ [0, 1),  ∈ B ⊥ if and only if  ∈ [0, 1) and e2π i·nk = 1 for all k ∈ Z. Hence n−1 1 }. B ⊥ = {0, , . . . , n n 1 We can take ΩB ⊥ = [0, ). Notice that as a subgroup of

R the annihilator of B n 1 is Z. n Example 2 Let p, q ∈ N, d = pq, and G = Zd = {0, 1, . . . , d − 1}. Let L = {0, p, 2p, . . . p(q − 1)} = {np : n = 0, . . . , q − 1} ≈ Zq . Its annihilator lattice is   λnp L⊥ = λ ∈ {0, 1, . . . , d − 1} : e2π i d = 1, ∀ n = 0, . . . , q − 1 = {0, q, 2q, . . . q(p − 1)} = {kq : k = 0, . . . , p − 1} ≈ Zp .

≈ Zd is ΩL⊥ = {0, . . . , q − 1} ≈ Zq . A fundamental set ΩL⊥ for L⊥ in G  ⊥ The characters ω ∈ L = {homomorphisms : L⊥ → T} of this group are of the 2π i λν

form (see, e.g., [8, Lemma 5.1.3]) ων (λ) = e p , λ ∈ L⊥ for ν ∈ { q :  = 0, . . . , p − 1} ≈ Zp . Suppose now that p = rs for some r, s ∈ N, and let B ⊂ L⊥ be B = {0, rq, 2rq, . . . , (s − 1)rq} = {j rq : j = 0, . . . , s − 1} ≈ Zs .

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D. Barbieri et al.

 ⊥ thus reads The annihilator of B in L    2π i bjprq B ⊥ = b ∈ { :  = 0, . . . , p − 1} : e = 1, ∀j = 0, . . . , s − 1 q s(r − 1) s s 2s } = {h : h = 0, . . . , r − 1} ≈ Zr . = {0, , , . . . , q q q q  ⊥ = {  :  = 0, . . . , p − 1} for B ⊥ is A fundamental set in L q  1 s − 1 ≈ Zs . ΩB ⊥ = 0, , . . . , q q By Proposition 3.3 in [7], the mapping K : 2 (L⊥ ) → L2 (ΩB ⊥ , 2 (B ⊥ )) given by K ({a(λ)}λ∈L⊥ )(t) = {({a(λ)}λ∈L⊥ )∧ (t + b)}b∈B ⊥ ⎧ ⎫ ⎨ ⎬ = a(λ)(t + b, λ) ⎩ ⊥ ⎭ λ∈L

(7)

b∈B ⊥

is an isometric isomorphism. Moreover, each B-invariant subspace J (ω), ω ∈ ΩL⊥ , has an associated measurable range function J (ω, ·) : ΩB ⊥ −→ {closed subspaces of 2 (B ⊥ )},  2 (B ⊥ )

such that for almost every t ∈ ΩB ⊥ , J (ω, t) = span {K (T ϕ)(ω))(t) : ϕ ∈ A } . From the definition of T given in (3) and the definition of K given in (7) we obtain K (T ϕ)(ω))(t) =

⎧ ⎨ ⎩

f (ω + λ)(t + b, λ)

λ∈L⊥

⎫ ⎬ ⎭

,

(8)

b∈B ⊥

when f ∈ L2 (G), ω ∈ ΩL⊥ , and t ∈ ΩB ⊥ .

and t ∈ G define For f ∈ L2 (G), ω ∈ G, Z f (ω, t) :=



f (ω + λ)(t, λ) ,

(9)

λ∈L⊥

the Zak transform of f with respect to the lattice L⊥ . Observe that in terms of this map, K (T ϕ)(ω))(t) = {Z f (ω, t + b)}b∈B ⊥ . To simplify the statement of the next theorem we write Xβ for the character on G associated with β ∈ B, that is, Xβ : G −→ T with Xβ (x) = (x, β) for all

Time-Frequency Data Approximation

35

associated with  ∈ L, that is, x ∈ G. Similarly X will denote the character on G

−→ T with X (ω) = (, ω) for all ω ∈ G.

X : G Theorem 2 Let G be a second countable LCA group, L and B be uniform lattices

respectively, with B ⊂ L⊥ . Let Γ = L × B and for f ∈ L2 (G), ω ∈ in G and G, ΩL⊥ , and t ∈ ΩB ⊥ define HΓ f (ω, t) = {Z f (ω, t + b)}b∈B ⊥ .

(10)

Then (1) The map HΓ intertwines Π , see (1), with the characters of Γ , that is, HΓ Π (, β)f = X− X−β HΓ f for all f ∈ L2 (G),  ∈ L, β ∈ B. (2) The map HΓ defined in (10) is an isometric isomorphism from L2 (G) onto L2 (ΩL⊥ × ΩB ⊥ , 2 (B ⊥ )). Proof For each b ∈ B ⊥ , the definition of Z given in (9) and the properties of the Fourier transform give Z Π (, β)f (ω, t + b) =



T  Mβ f (ω + λ)(t + b, λ)

λ∈Λ⊥

=



(, ω + λ)f (ω + λ − β)(t + b, λ) .

λ∈Λ⊥

Using that (, λ) = 1 and the change of variables λ − β = λ ∈ L⊥ yields Z Π (, β)f (ω, t + b) = (, ω)



f (ω + λ )(t + b, λ + β) .

λ ∈Λ⊥

Using that (t + b, β) = (t, β) · (b, β) = (t, β) we obtain Z Π (, β)f (ω, t + b) = (, ω) (t, β)



f (ω + λ )(t + b, λ )

λ ∈Λ⊥

= X− (ω)X−β (t)Z f (ω, t + b) . This proves (1). To prove (2) observe that by the definition of HΓ given in (10) together with (8) and (9) we have HΓ f (ω, t) = K (T f (ω))(t) . That HΓ is an isometry now follows from the fact that T and K are isometries in their respective spaces. We need to prove that HΓ is onto. Since K : 2 (L⊥ ) → L2 (ΩB ⊥ , 2 (B ⊥ )) is an isometric isomorphism between Hilbert spaces, by Lemma 1 in the Appendix, the map

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D. Barbieri et al.

QK : L2 (ΩL⊥ , 2 (L⊥ )) −→ L2 (ΩL⊥ , L2 (ΩB ⊥ , 2 (B ⊥ )) given by (QK f )(ω) = K (f (ω)), f ∈ L2 (ΩL⊥ , 2 (L⊥ )) is an isometric isomorphism. Moreover, by Fubini’s theorem, the Hilbert spaces L2 (ΩL⊥ , L2 (ΩB ⊥ , 2 (B ⊥ )) and L2 (ΩL⊥ × ΩB ⊥ , L2 (2 (B ⊥ )) are also isomorphic and the isomorphism is given by Φ(f )(ω, t) = f (ω)(t) , for f ∈ L2 (ΩL⊥ , L2 (ΩB ⊥ , 2 (B ⊥ )) . Let now F ∈ L2 (ΩL⊥ × ΩB ⊥ , L2 (2 (B ⊥ )). Choose g ∈ L2 (ΩL⊥ , 2 (L⊥ )) such that Φ ◦ QK (g) = F. Hence F (ω, t) = Φ ◦ QK (g)(ω, t) = QK (g)(ω)(t) = K (g(ω))(t). Choose now f ∈ L2 (G) such that T (f ) = g. Then HΓ f (ω, t) = K (T f (ω))(t) = F (ω, t).  

This finishes the proof of the theorem. Example 3 For the cyclic group of Example 2, recall that, for f ∈ Cd d−1 gω 1  f (g)e−2π i d , ω ∈ {0, . . . , d − 1}. f (ω) = √ d g=0

  , the Zak transform (9) thus reads For t ∈ ΩB ⊥ = 0, q1 , . . . , s−1 q Z f (ω, t) =

p−1 

−2π i f (ω + kq)e

kqt p

k=0

=

p−1  k=0

d−1 g(ω+kq) −2π i kqt 1  p f (g)e−2π i d e √ d g=0

d−1 d−1 gω gω 1  e2π i d  =√ f (g)e−2π i d K(g + qt) = √ f (g − qt)e−2π i d K(g), d g=0 d g=0 qtω

where K(g) =

p−1 

e

 −2π i pg k

=

k=0

Z f (ω, t) =

√

pe2π i

p if g ∈ L . This gives 0 if g ∈ /L qtω d

q−1 pnω 1  f (pn − qt)e−2π i d . √ q n=0

Time-Frequency Data Approximation

37

Before embarking in the proof of Theorem 1, which will be accomplished in Sect. 3, we need an additional result. Let V = SΓ (A ) be a Γ -invariant subspace of L2 (G), where A ⊂ L2 (G). For each (ω, t) ∈ ΩL⊥ × ΩB ⊥ , consider the range function JV : ΩL⊥ × ΩB ⊥ −→ {closed subspaces of 2 (B ⊥ )} given by  2 (B ⊥ )

JV (ω, t) := span {HΓ ϕ(ω, t) : ϕ ∈ A }

.

(11)

Proposition 1 With V = SΓ (A ) as above, let PJV (ω,t) be the orthogonal projection of 2 (B ⊥ ) onto JV (ω, t). Then, for all f ∈ L2 (G) and (ω, t) ∈ ΩL⊥ × ΩB ⊥ , H PSΓ (A ) f (ω, t) = PJV (ω,t) (HΓ f (ω, t)) . Proof Observe first that, since HΓ is an isometric isomorphism between Hilbert spaces, then HΓ PSΓ (A ) = PHΓ (SΓ (A )) HΓ .

(12)

The set D := {X Xβ : (, β) ∈ Γ } of characters of Γ is a determining set for L1 (ΩL⊥ × ΩB ⊥ ) in the sense of Definition 2.2 in [5], because  ΩL⊥ ×ΩB⊥

f (ω, t)X (ω)Xβ (t)dωdt = 0 ⇒ f = 0,

∀ f ∈ L1 (ΩL⊥ × ΩB ⊥ ).

Indeed, this is Fourier uniqueness theorem since ΩL⊥ and ΩB ⊥ are relatively compact. By (1) of Theorem 2, for all f ∈ L2 (G), HΓ (T Mβ f ) = X− X−β (HΓ f ). Thus, HΓ (SΓ (A )) is D-multiplicative invariant in the sense of Definition 2.3 in [5]. Indeed, if X Xβ ∈ D, F ∈ HΓ (SΓ (A )) and writing HΓ f = F we have X Xβ F = X Xβ (HΓ f ) = HΓ (T− M−β f ) ∈ HΓ (SΓ (A )) . By Theorem 2.4 in [5], JV is a measurable range function. By Proposition 2.2 in [5], PHΓ (SΓ (A )) (HΓ f )(w, t) = PJV (ω,t) (HΓ f (ω, t)) . The result now follows from (12).

 

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3 Solution to the Approximation Problem This section is dedicated to the proof of Theorem 1. Let F = {f1 , . . . , fm } ⊂ L2 (G) be a collection of functional data. With the notation of Theorem 1, for each n < m we need to find {ψ1 , . . . , ψn } ⊂ L2 (G) such that E (F ; SΓ {ψ1 , . . . , ψn }) ≤ E (F ; V ) for any Γ -invariant subspace V of L2 (G) of length less than or equal to n. The definition of E (F ; V ) is given in (2) and for convenience of the reader we recall it here. E (F ; V ) :=

m  j =1

fj − PV fj 2L2 (G) .

For a.e. (ω, t) ∈ ΩL⊥ × ΩB ⊥ consider HΓ (F )(w, t) := {HΓ f1 (ω, t), . . . , HΓ fm (ω, t)} . Let GF ,Γ (w, t) be the m × m C-valued matrix whose (i, j ) entry is given by [GF ,Γ (w, t)]i,j = HΓ fi (ω, t), HΓ fj (ω, t) 2 (B ⊥ ) . The matrix GF ,Γ (w, t) is Hermitian and its entries are measurable functions defined on ΩL⊥ × ΩB ⊥ . Write λ1 (ω, t) ≥ λ2 (ω, t) ≥ . . . , ≥ λm (ω, t) ≥ 0 for the eigenvalues of GF ,Γ (w, t). By Lemma 2.3.5 in [10] the eigenvalues λi (ω, t), i = 1, . . . , m, are measurable and there exist corresponding measurable vectors yi (ω, t) = (yi,1 (ω, t), . . . , yi,m (ω, t)) that are orthonormal left eigenvectors of the matrix GF ,Γ (w, t). That is, yi (ω, t) GF ,Γ (w, t) = λi (ω, t) yi (ω, t),

i = 1, . . . , m.

(13)

For n ≤ m, define q1 (ω, t), . . . , qn (ω, t) ∈ 2 (B ⊥ ) by σi (ω, t) qi (ω, t) = 

m 

yi,j (ω, t) HΓ fj (ω, t)

i = 1, . . . , n,

j =1

where   σi (ω, t) =

√ 1 λi (ω,t)

0

if λi (ω, t) = 0 otherwise.

(14)

Time-Frequency Data Approximation

39

By the Eckart–Young Theorem (see the version stated and proved in Theorem 4.1 of [1]), {q1 (ω, t), . . . , qn (ω, t)} is a Parseval frame for the space it generates Q(ω, t) := span {q1 (ω, t), . . . , qn (ω, t)} and Q(ω, t) is optimal in the sense that E(HΓ (F )(w, t); Q(ω, t)) :=

m 

HΓ fi (ω, t) − PQ(ω,t) HΓ (fi )(w, t) 22 (B ⊥ )

i=1

≤

m 

HΓ fi (ω, t) − PQ HΓ (F )(w, t) 22 (B ⊥ ) := E(HΓ (fi )(w, t); Q )

i=1

(15) for any Q subspace of 2 (B ⊥ ) of dimension less than or equal to n. Moreover, E(HΓ (F )(w, t); Q(ω, t)) =

m 

λi (ω, t).

(16)

i=n+1

Before continuing with the proof, let us relate the pointwise errors that appear in (15) to the error defined in (2) for Γ -invariant subspaces. Proposition 2 For V = SΓ (A ) as in Proposition 1,   E(HΓ (F )(w, t); JV (ω, t)) dtdω , E (F ; V ) = ΩL⊥

Ω B⊥

where JV (ω, t) is defined in (11). Proof By (2) of Theorem 2, HΓ is an isometry from L2 (G) onto the space L2 (ΩL⊥ × ΩB ⊥ , 2 (B ⊥ )). Therefore, E (F ; V ) =

m  j =1

=

m  j =1

=

fj − PV fj 2L2 (G)

HΓ fj − HΓ PV fj 2L2 (Ω

m  

L⊥ ×ΩB⊥ ,

2 (B ⊥ ))



j =1 ΩL⊥

Ω B⊥

HΓ fj (ω, t) − HΓ PV fj (ω, t) 22 (B ⊥ ) dtdω .

By Proposition 1, 



m 

E (F ; V ) = 

ΩL⊥



ΩB⊥ j =1

=

HΓ fj (ω, t) − PJV (ω,t) (HΓ fj (ω, t)) 22 (B ⊥ ) dtdω

E(HΓ (F )(w, t); JV (ω, t)) dtdω . ΩL⊥

Ω B⊥

 

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D. Barbieri et al.

Let us now continue with the proof of Theorem 1. By definition (14), each qi (ω, t) is measurable and defined on ΩL⊥ × ΩB ⊥ with values in 2 (B ⊥ ). Moreover,

qi (ω, t) 22 (B ⊥ ) = qi (ω, t), qi (ω, t) 2 (B ⊥ ) = σi (ω, t)2

m m   

yi,j (ω, t) Z fj (ω, t + b) Z fs (ω, t + b) yi,s (ω, t)

b∈B ⊥ j =1 s=1

= σi (ω, t)2

m 

yi,j (ω, t)

j =1

m  Zfj (ω, t), Zfs (ω, t) 2 (B ⊥ ) yi,s (ω, t) . s=1

In matrix form, t

qi (ω, t) 22 (B ⊥ ) =  σi (ω, t)2 yi (ω, t) GF ,Γ (w, t) yi (ω, t) . By (13), the orthonormality of the vectors yi (ω, t), and the definition of  σi (ω, t), we have σi (ω, t)2 λi (ω, t) yi (ω, t) 2 ≤ 1 .

qi (ω, t) 22 (B ⊥ ) =  Since ΩL⊥ and ΩB ⊥ have finite measure, we conclude that for i = 1, . . . , n, qi ∈ L2 (ΩL⊥ × ΩB ⊥ , 2 (B ⊥ )). The mapping HΓ is onto by part (2) of Theorem 2. Therefore there exist ψi ∈ L2 (G) such that HΓ (ψi ) = qi ,

i = 1, . . . , n.

It remains to show that the space W := SΓ (ψ1 , . . . , ψn ) is the optimal one as required in the statement of Theorem 1. By Proposition 2 

 E (F ; W ) =

E(HΓ (F )(w, t); JW (ω, t)) dtdω . ΩL⊥

Ω B⊥

By (15) and the definitions of ψi , JW (ω, t) = Q(ω, t). Therefore, we can write, 

 E (F ; W ) =

E(HΓ (F )(w, t); Q(ω, t)) dtdω . ΩL⊥

(17)

Ω B⊥

Let now V = SΓ (ϕ1 , . . . , ϕr ), r ≤ n, be any Γ -invariant subspace of length less than or equal n. Since JV (ω, t) has dimension less than or equal n, (15) gives 



E(HΓ (F )(w, t); JV (ω, t)) dtdω = E (F ; V ) ,

E (F ; W ) ≤ ΩL⊥

Ω B⊥

Time-Frequency Data Approximation

41

where the last equality is due to Proposition 2. Moreover, by (17) and (16) m  

E (F ; W ) =

i=n+1 ΩL⊥

 λi (ω, t)dωdt. Ω B⊥

This finishes the proof of Theorem 1.



Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777822. In addition, D. Barbieri and E. Hernández were supported by Grant MTM2016-76566-P (Ministerio de Ciencia, Innovación y Universidades, Spain). C. Cabrelli and U. Molter were supported by Grants UBACyT 20020170100430BA (University of Buenos Aires), PIP11220150100355 (CONICET) and PICT 2014-1480 (Ministerio de Ciencia, Tecnología e Innovación, Argentina).

Appendix We give the proof of the following Lemma that has been used in Sect. 2 to prove part (2) of Theorem 2. Lemma 1 Let σ : H1 −→ H2 be an isometric isomorphism between the Hilbert spaces H1 and H2 . For a measure space (X, dμ) the map Qσ : L2 (X, H1 ) −→ L2 (X, H2 ) given by (Qσ f )(x) = σ (f (x)) is also an isometric isomorphism. Proof Let f be a measurable vector function in L2 (X, H1 ), that is, for every y ∈ H1 the scalar function x −→ f (x), y H1 is measurable. We must prove that Qσ f is also a measurable vector function in L2 (X, H2 ). For z ∈ H2 we have < Qσ f (x), z >H2 =< σ (f (x)), z >H2 =< f (x), σ ∗ (z) >H1 . Since σ ∗ (z) = σ −1 (z) is a general element of H1 , this shows that Qσ f is measurable. Moreover, for f, g ∈ L2 (X, H1 ),  < Qσ f, Qσ g >L2 (X,H2 ) =

X

< σ (f (x)), σ (g(x)) >H2 dμ(x)

 =

X

< f (x), g(x) >H1 dμ(x) =< f, g >L2 (X,H1 ) .

This shows that if f ∈ L2 (X, H1 ), Qσ f ∈ L2 (X, H2 ) and that Qσ is an isometry. Finally, it is easy to see that R : L2 (X, H2 ) → L2 (X, H1 ) defined by Rg(x) = −1   σ (g(x)) is the inverse and the adjoint of Qσ . Therefore, Qσ is onto.

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References 1. A. Aldroubi, C. Cabrelli, D. Hardin, U. Molter, Optimal shift invariant spaces and their Parseval frame generators. Appl. Comput. Harmon. Anal. 23 (2007), pp. 273–283. 2. D. Barbieri, C. Cabrelli, E. Hernández, U. Molter, Approximation by group invariant subspaces. J. Math. Pure. Appl. (2020) (in press). https://doi.org/10.1016/j.matpur.2020.08.010 3. D. Barbieri, E. Hernández, V. Paternostro, Spaces invariant under unitary representations of discrete groups. J. Math. Anal. Appl. (2020) (in press). https://doi.org/10.1016/j.jmaa.2020. 124357 4. C. de Boor, R. A. DeVore, A. Ron, Approximation from shift-invariant subspaces of L2 (Rd ). Trans. Amer. Math. Soc. 341 (1994), pp. 787–806. 5. M. Bownik, K. Ross, The structure of translation-invariant spaces on locally compact abelian groups. J. Fourier Anal. Appl. 21 (2015), pp. 849–884. 6. C. Cabrelli, C. Mosquera, V. Paternostro, An approximation problem in multiplicatively invariant spaces. In “Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth”, M. Cwikel and M. Milman (eds.). Contemp. Math. 693 (2017), pp. 143–166. 7. C. Cabrelli, V. Paternostro, Shift-invariant spaces on LCA groups. J. Funct. Anal. 258 (2010), pp. 2034–2059. 8. A. Deitmar, A first course in harmonic analysis. Springer, 2nd ed. 2005. 9. K.-H. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, (2001). 10. A. Ron, Z. Shen, Frames and stable bases for shift-invariant subspaces of L2 (Rd ), Canad. J. Math., 47, (1995), no. 5, 1051-1094. 11. W. Rudin, Fourier Analysis on Groups, John Wiley, (1992).

The Shearlet Transform and Lizorkin Spaces Francesca Bartolucci, Stevan Pilipovi´c, and Nenad Teofanov

Abstract We prove a continuity result for the shearlet transform when restricted to the space of smooth and rapidly decreasing functions with all vanishing moments. We define the dual shearlet transform, called here the shearlet synthesis operator, and we prove its continuity on the space of smooth and rapidly decreasing functions over R2 × R × R× . Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions via the duality approach, and we prove its consistency with the classical definition for test functions and its equivalence with the coorbit space approach. Keywords Shearlet transform · Wavelet transform · Radon transform · Ridgelet transform · Lizorkin spaces

1 Introduction Among the large reservoir of directional multiscale representations which have been introduced over the years, the shearlet representation has gained considerable attention for its capability to resolve the wavefront set of distributions, providing both the location and the geometry of the singularity set of signals. Indeed, when we shift from one-dimensional to multidimensional signals, it is not just of interest to locate singularities in space but also to describe how they are distributed. This additional information is expressed by the notion of wavefront set introduced by Hörmander in [14]. In [16] the authors show that the decay rate of the shearlet coefficients Sψ f (b, s, a) of a signal f with respect to suitable shearlets

F. Bartolucci () Department of Mathematics, ETH Zurich, Zurich, Switzerland e-mail: [email protected] S. Pilipovi´c · N. Teofanov Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_3

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ψ characterizes the wavefront set of f . Precisely, they show that for any signal f ∈ L2 (R2 ) the shearlet coefficients Sψ f (b, s, a) exhibit fast asymptotic decay as a → 0 except when the pair (b, (ξ1 , ξ2 )) ∈ R2 × R2 , with ξ2 /ξ1 = s, belongs to the wavefront set of f . Later this result has been generalized in [10] where it is shown that the same result holds true under much weaker assumptions on the admissible vectors by means of a new approach based on an adaptation of the Radon transform to the shearlet structure, the affine Radon transform. On the other hand, while the classical wavelet transform is widely exploited in signal analysis for describing pointwise smoothness of univariate functions (we refer to [15, 18] as classical references), it has proved not flexible enough to capture the geometry of the singularity set when we shift from one-dimensional signals to multidimensional signals. We refer to, e.g., [20] for a modification of the wavelet transform which overcomes these difficulties. In some sense, shearlets behave for high-dimensional signals as wavelets do for one-dimensional signals and the link between these two transforms has been clarified in [2] where it is shown that the shearlet transform is the composition of the affine Radon transform with a one-dimensional wavelet transform, followed by a convolution operator with a scale-dependent filter. There are at least two classical approaches to extend integral transforms to generalized function spaces. The coorbit space theory introduced by Feichtinger and Gröchenig in [6, 7] applies when the integral transform is the voice transform associated with a square-integrable representation of a locally compact group, and this is the case of the shearlet transform. We refer to [5] for an extension of the shearlet transform based on the coorbit space theory. The second way to proceed is the duality approach introduced by Schwartz in the 50’s. A classic example is the extension of the Fourier transform to the space of tempered distributions. In this paper we extend the shearlet transform to distributions following the approach of Schwartz. Our work arises from the lack of a complete distributional framework for the shearlet transform in the literature and from the link between the shearlet transform with the Radon and the wavelet transforms, whose distribution theory is deeply investigated and a well-known subject in applied mathematics. We refer, respectively, to [13, 21] and to [11, 12] for the extension of the wavelet transform and the Radon transform to various generalized function spaces via a duality approach. The Lizorkin space plays a crucial role in the development of a distributional framework for these two classical transforms and it turns out to be a natural domain for the shearlet transform too. We recall that the Lizorkin space S0 (Rd ) consists of smooth and rapidly decreasing functions with vanishing moments of any order. Moreover, in [19] the authors show that the domain of the ridgelet transform can be enlarged to its dual space S0 (R2 ), known as the space of Lizorkin distributions. Their proofs widely exploit the intimate connection between the Radon, the ridgelet and the wavelet transforms, which also yields a relation formula between the shearlet transform and the ridgelet transform as we show in the Appendix in Proposition 3. This has in part inspired our work and we adapt several ideas of [19] to our context.

The Shearlet Transform and Lizorkin Spaces

45

Our main results are continuity theorems for the shearlet transform and its dual transform, called the shearlet synthesis operator, on various test function spaces. Precisely, we prove that the shearlet transform Sψ : S0 (R2 ) → S (S) and its dual transform Sψt : S (S) → S (R2 ) are continuous, where S (S) is a certain space of highly localized functions (see Sect. 2.1 for the definition of S (S)). Our continuity theorems hold for suitable choice of the admissible vector ψ in the space S0 (R2 ) (see Sect. 3) and this is not a surprising condition. Indeed, as pointed out in wavelet analysis [18], shearlet analysis [10] and in the study of the Taylorlet transform [8] vanishing moments are crucial in order to measure the local regularity and to detect anisotropic structures of a signal. Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions following the approach in [19]. We show that the shearlet transform can be extended as a continuous map from S0 (R2 ) into S  (S), where S  (S) is the space of distributions of slow growth on R2 × R × R× . Observe that many important Schwartz distribution spaces, such as E  (Rd ), OC (Rd ), Lp (Rd ), and DL 1 (Rd ), are embedded into the space of Lizorkin distributions S0 (Rd ) (see, e.g., [19]). When considering possible applications of our approach, we notice that the rectified linear units (ReLUs), which are important examples of unbounded activation functions in the context of deep learning neural networks, belong to the space of Lizorkin distributions, see [23] for details. The chapter is organized as follows. In Sect. 2 we introduce the spaces that occur in our analysis. Then, we recall the definition and the basic properties of the wavelet transform and the Radon transform in polar and affine coordinates. Section 3 is devoted to an introduction of the shearlet transform and to recall one of the main results in [2]. Moreover, in Theorem 2, we give a sketch of the proof of the continuity of the shearlet transform Sψ on S0 (R2 ). In Sect. 4 we introduce and study the shearlet synthesis operator. In particular, in Theorem 3 we prove its continuity on S (S). The importance of this dual transform follows by the fact that it can be used to define the extension of the shearlet transform to the space of Lizorkin distributions S0 (R2 ) in a natural way, as we show in Sect. 5. We conclude our analysis with Theorem 4 which proves that our definition of the shearlet transform of distributions extends the ones considered so far, see, e.g., [10, 17], and it is consistent with those for test functions (see Definition 1). Moreover, Theorem 4 shows that our duality approach is equivalent to the one based on the coorbit space theory presented in [5].

1.1 Notation We briefly introduce the notation. We set N = {0, 1, 2, . . . }, Z+ denotes the set of positive integers, R+ = (0, +∞), R× = R\{0} and Hd+1 = Rd ×R× , d ∈ Z+ . We also use the notation H(m1 ,...,md ,1) = Rm1 ×. . .×Rmd ×R× , mj ∈ Z+ , j = 1, . . . , d. When x, y ∈ Rd and m ∈ Nd , |x| denotes the Euclidean norm, x · y their scalar product, x = (1 + |x|2 )1/2 , xy = x1 y1 + x2 y2 + · · · + xd yd , x m = x1m1 . . . xdmd and

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∂ m = ∂xm = ∂xm11 . . . ∂xmdd . We write also ϕ (m) = ∂ m ϕ, m ∈ Nd . By a slight abuse of notation, the length of a multi-index m ∈ Nd is denoted by |m| = m1 +· · ·+md and the meaning of | · | shall be clear from the context. We write A  B when A ≤ C · B for some positive constant C. For any p ∈ [1, +∞] we denote by Lp (Rd ) the Banach space of functions f : Rd → C that are p-integrable with respect to the Lebesgue measure dx and, if p = 2, the corresponding scalar product and norm are ·, · and · , respectively. The Fourier transform is denoted by F both on L2 (Rd ) and on L1 (Rd ), where it is defined by  F f (ξ ) =

Rd

f (x)e−2π i ξ ·x dx,

f ∈ L1 (Rd ).

If G is a locally compact group, we denote by L2 (G) the Hilbert space of squareintegrable functions with respect to a left Haar measure on G, and C(G) denotes the space of continuous functions on G. If A ∈ Md (R), the vector space of square d × d matrices with real entries, tA denotes its transpose and we denote the (real) general linear group of size d × d by GL(d, R). Finally, for every b ∈ Rd , the translation operator acts on a function f : Rd → C as Tb f (x) = f (x − b) and the dilation 1 operator Da : Lp (Rd ) → Lp (Rd ) is defined by Da f (x) = |a|− 2 f (x/a) for every a ∈ R× . The dual pairing between a test function space A and its dual space of distributions A  is denoted by ( · , · ) = A  ( · , · )A and we provide all distribution spaces with the strong dual topologies. The Schwartz space of rapidly decreasing smooth test functions is denoted by S (Rd ) and S  (Rd ) denotes its dual space of tempered distributions. For the seminorms on S (Rd ), we make the choice ρν (ϕ) =

sup

x ν |∂ m ϕ(x)|,

x∈Rd ,|m|≤ν

for every ν ∈ N and ϕ ∈ S (Rd ).

2 Preliminaries In this section we first introduce the Lizorkin space of test functions, an important subspace of S (Rd ) which plays a crucial role in our analysis. Afterwards, we recall the definition and the main properties of the wavelet transform and the Radon transform in polar and affine coordinates in order to recall in Sect. 3 part of the results contained in [2].

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47

2.1 The Spaces In this subsection we introduce the spaces that occur in this chapter and we state some auxiliary results which we widely exploit in the proofs of our main results (see Lemmas 1 and 2 below). In particular, the Lizorkin space S0 (Rd ) will play a crucial role in our analysis. It consists of rapidly decreasing functions with vanishing moments of any order. Precisely,   S0 (Rd ) = ϕ ∈ S (Rd ) : μm (ϕ) = 0, ∀m ∈ Nd ,  where μm (ϕ) = Rd x m ϕ(x)dx, m ∈ Nd . The Lizorkin space S0 (Rd ) is a closed subspace of S (Rd ) equipped with the relative topology inhered from S (Rd ) and its dual space of Lizorkin distributions S0 (Rd ) is canonically isomorphic to the quotient of S  (Rd ) by the space of polynomials (cf. [13, 19]). We are also interested in the Fourier Lizorkin space Sˆ0 (Rd ) which consists of rapidly decreasing functions that vanish in zero together with all their partial derivatives, i.e.,   Sˆ0 (Rd ) = ϕ ∈ S (Rd ) : ∂ m ϕ(0) = 0, ∀m ∈ Nd , which is a closed subspace of S (Rd ) too and we endow it with the relative topology inhered from S (Rd ). We observe that, since S0 (Rd ) and Sˆ0 (Rd ) are closed subspaces of the nuclear ˆ the topological tensor space S (Rd ), they are nuclear as well. We denote by X⊗Y product space obtained as the completion of X ⊗ Y in the inductive tensor product topology ε or the projective tensor product topology π , see [24] for details. Then, we have the following result. Lemma 1 The spaces S0 (Rd ) and Sˆ0 (Rd ) are closed under translations, dilations, differentiations, and multiplications by a polynomial. Moreover, the Fourier transform is an isomorphism between S0 (Rd ) and Sˆ0 (Rd ) and we have the following canonical isomorphisms: ˆ 0 (Rd2 ), S0 (Rd ) ∼ = S0 (Rd1 )⊗S ˆ Sˆ0 (Rd2 ), Sˆ0 (Rd ) ∼ = Sˆ0 (Rd1 )⊗ ˆ denotes the completion with respect to the where d = d1 + d2 ∈ Z+ , and ⊗ ε−topology or the π -topology. Proof The proof is based on classical arguments and we omit it (cf. [24, Theorem 51.6] for the canonical isomorphisms).   The next Lemma is a reformulation of [18, Theorem 6.2].

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Lemma 2 ([1]) Let f ∈ S0 (Rd ). Then, for any given m ∈ Nd there exists g ∈ S0 (Rd ) such that F f (ξ ) = ξ m F g(ξ ),

ξ ∈ Rd ,

and vice versa. Proof We start proving the above statement for d = 1 and m = 1. Let f ∈ S0 (R) and consider  x  +∞ g(x) = f (t)dt = − f (t)dt, −∞

x

where in the second equality we use the fact that f ∈ S0 (R). For every k ∈ N and x>0 ! +∞ !  +∞ ! ! k k 2 k2 ! (1 + x ) f (t)dt !! ≤ (1 + t 2 ) 2 |f (t)|dt x |g(x)| = ! x

x



≤ ρ2k+4 (f )

+∞ −∞

k

(1 + t 2 ) 2

1 dt < +∞. (1 + t 2 )k+2

Analogously, for x < 0 it holds ! ! x k |g(x)| = !!

x

−∞

!  ! k (1 + x 2 ) 2 f (t)dt !! ≤ 

≤ ρ2k+4 (f )

+∞ −∞

k

(1 + t 2 ) 2

x

k

−∞

(1 + t 2 ) 2 |f (t)|dt

1 dt < +∞. (1 + t 2 )k+2

Thus, g is a well-defined function and supx∈R x k |g(x)| < +∞ for every k ∈ N. Moreover, g  (x) = f (x), so that supx∈R x k |g (l) (x)| < +∞ for every k ∈ N and l ≥ 1. Therefore, g ∈ S (R). Furthermore, for any n ∈ N, we have that 

+∞ −∞

 x n g(x)dx = −

+∞ −∞

x n+1 g  (x)dx = −



+∞ −∞

x n+1 f (x)dx = 0.

Hence, g ∈ S0 (R) and by the definition of g we have F f (ξ ) = F g  (ξ ) = (2π i)ξ F g(ξ ),

ξ ∈ R.

The opposite direction is obviously true since the space S0 (R) is closed under multiplication by a polynomial and this concludes the proof for d = 1 and m = 1. The analogous statement holds true for m > 1 by iterating the above proof m-times. The case d > 1 follows by analogous computations.  

The Shearlet Transform and Lizorkin Spaces

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By Lemma 2, if f ∈ S0 (R2 ), then for any given k, l ∈ N there exists g ∈ S0 (R2 ) such that F f (ξ1 , ξ2 ) = ξ1k ξ2l F g(ξ1 , ξ2 ),

(ξ1 , ξ2 ) ∈ R2 .

Moreover, it is worth observing that by Lemmas 1 and 2, f ∈ S0 (Rd ) if and only if it satisfies the directional vanishing moments:  xjm f (x1 , x2 , . . . , xd )dxj = 0, ∀m ∈ N, j = 1, . . . , d. R

The space S (H(d,d−1,1) ) of highly localized functions (see also [13]) consists of the functions Φ ∈ C ∞ (H(d,d−1,1) ) such that the seminorms α ,α ,β,γ

ρk11,k22,l,m (Φ) = ˜ k2 s l b1 b



k1

sup (b,s,a)∈H(d,d−1,1)

1 |a| + m |a| m

! ! ! γ β α2 α1 ! !∂a ∂s ∂b˜ ∂b1 Φ(b, s, a)!

˜ ∈ are finite for all k1 , m, α1 , γ ∈ N, k2 , l, α2 , β ∈ Nd−1 and where b = (b1 , b) R × Rd−1 . In particular, when d = 2, we denote S := H(d,d−1,1) = R2 × R × R× and S (S) consists of the functions Φ ∈ C ∞ (S) such that the seminorms α ,α ,β,γ

pk11,k22,l,m (Φ)

 ! ! 1 ! γ ! m = sup b1 b2 s |a| + m !∂a ∂sβ ∂bα22 ∂bα11 Φ((b1 , b2 ), s, a)! |a| ((b1 ,b2 ),s,a)∈S (1) k1

k2

l

are finite for all k1 , k2 , l, m, α1 , α2 , β, γ ∈ N. The topology of S (S) is defined by means of the seminorms (1). Its dual S  (S) will play a crucial role in the definition of the shearlet transform of Lizorkin distributions since it contains the range of this transform. We fix dμ(b, s, a) = |a|−3 dbdsda as the standard measure on S, where db, ds, and da are the Lebesgue measures on R2 , R, and R× , respectively. If F is a function of at most polynomial growth on S, i.e., if there exist C, ν1 , ν2 , ν3 > 0 such that   1 (b, s, a) ∈ S, |F (b, s, a)| ≤ Cb ν1 s ν2 |a|ν3 + ν , |a| 3 then we identify F with an element of S  (S) by means of the equality  (F, Φ) = for every Φ ∈ S (S).

  R×

R R2

F (b, s, a)Φ(b, s, a)

dbdsda , |a|3

(2)

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2.2 The Wavelet Transform The one-dimensional affine group W is the semidirect product R  R× with group operation (b, a)(b , a  ) = (b + ab , aa  ) and left Haar measure |a|−2 dbda. It acts on L2 (R) by means of the square-integrable representation 1

Wb,a f (x) = |a|− 2 f



 x−b , a

or, equivalently, in the frequency domain 1

F Wb,a f (ξ ) = |a| 2 e−2π ibξ F f (aξ ).

(3)

The wavelet transform is then Wψ f (b, a) = f, Wb,a ψ , which is a multiple of an isometry from L2 (R) into L2 (W, |a|−2 dbda) provided that ψ ∈ L2 (R) satisfies the admissibility condition, namely the Calderón equation,  0
", with " > 0, we obtain that 



R2

R×

 R

k4 13 k |a|− 4 +k3 − 2 +m (1 + |ξ |2 )N/2 (1 + s 2 )|∂ξ 1 F Φ(ξ, s, a)||ξ1 |−k2 −k4 +m × 1

!  ! ! (k ) ξ2 /ξ1 − s ! dsda ! dξ1 dξ2 × × |F g(−aξ1 )||ξ2 |k4 !!φ2 4 ! 1/2 2 |a| (1 + s )(1 + |ξ |2 )N/2  [ρ

k1 ,0,0,0 k ,0,0,0 (F Φ)+ρ 1 (F Φ)], k4 k N +m−k2 −k4 ,N +k4 ,2,| 13 N +m−k2 −k4 ,N +k4 ,2,|−k3 + 24 −m| −k + −m| 3 4 2

which is dominated by a single seminorm. We can treat |x2 k (Sψt Φ)(x1 , x2 )|, k ∈ Z+ , in the same manner and we conclude that the shearlet synthesis Sψt is a continuous map from S (S) into S (R2 ). Finally, it remains to prove that Sψt Φ ∈ S0 (R2 ). The idea is to prove the equivalent condition lim

ξ →0

F Sψt Φ(ξ ) |ξ |k

= 0,

for every k ∈ N, see [13, Lemma 6.0.4]. We refer to [1] for the details.

 

5 The Shearlet Transform on S0 (R2 ) In this last section, we extend the definition of the shearlet transform to the space of Lizorkin distributions and we show that our definition extends the ones introduced so far. In particular, we prove its consistency with the classical definition for test functions. We recall that we consider admissible vectors ψ of the form (11) with χ1 defined by (13) in S0 (R) and φ2 = F ψ2 ∈ S (R). Definition 2 ([1]) We define the shearlet transform of f ∈ S0 (R2 ) with respect to ψ as follows (Sψ f, Φ) = (f, Sψt Φ),

Φ ∈ S (S).

The consistency of Definition 2 is guaranteed by Theorem 3. Furthermore, it follows straightforwardly that the shearlet transform of f ∈ S0 (R2 ) is a well-defined distribution in S  (S). Proposition 2 ([1]) The shearlet transform Sψ given by Definition 2 is a continuous and linear map from S0 (R2 ) into S  (S).

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The next theorem shows that Definition 2 is in fact consistent with the definition for test functions (Definition 1) and that it generalizes the extension considered in [10, 16] where the shearlet transform of a tempered distribution f with respect to an admissible vector ψ ∈ S (R2 ) is given by the function Sψ f (b, s, a) = S  (R2 ) (f, Sb,s,a ψ)S (R2 ) , for every (b, s, a) ∈ S. Following the coorbit space approach, given a suitable test function space usually denoted by H1,w , where w is a weight function, and its anti∼ , the extended shearlet transform of f ∈ H ∼ with respect to ψ ∈ H dual H1,w 1,w 1,w is defined by ∼ (f, Sb,s,a ψ)H , Sψ f (b, s, a) = H1,w 1,w

for every (b, s, a) ∈ S, [5]. Theorem 4 shows the equivalence of our duality approach with the coorbit space one. Precisely, Theorem 4 states that the shearlet transform of any Lizorkin distribution is given by the function defined as (b, s, a) → S  (R2 ) (f, Sb,s,a ψ)S0 (R2 ) , 0

for every (b, s, a) ∈ S. Theorem 4 ([1]) Let f ∈ S0 (R2 ). The shearlet transform of f is given by the function (b, s, a) → S  (R2 ) (f, Sb,s,a ψ)S0 (R2 ) , 0

that is,  (Sψ f, Φ) =

S

(f, Sb,s,a ψ)Φ(b, s, a)dμ(b, s, a),

Φ ∈ S (S).

Proof Consider f ∈ S0 (R2 ). Since the space of Lizorkin distributions S0 (R2 ) is canonically isomorphic to the quotient of S  (R2 ) by the space of polynomials, by Schwartz’ structural theorem [22, Theorém VI], we can write f = g (α) + p, where g is a continuous slowly growing function, α ∈ N2 and p is a polynomial. Then, for any Φ ∈ S (S) (g

(α)

, Sψt Φ)

|α|

= (−1)

= (−1)|α|  =

S

(g, Sψt Φ (α) )

= (−1)



 R2

g(x)

S

|α|

 R2

g(x)Sψt Φ (α) (x)dx

Φ(b, s, a) (Sb,s,a ψ)(α) (x) dμ(b, s, a)dx

Φ(b, s, a) (−1)|α|

 R2

g(x)(Sb,s,a ψ)(α) (x)dxdμ(b, s, a)

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 =

S

Φ(b, s, a) (−1)|α| (g, (Sb,s,a ψ)(α) )dμ(b, s, a)

 =

S

Φ(b, s, a) (g (α) , Sb,s,a ψ)dμ(b, s, a).

Analogously, we have that  (p, Sψt Φ) =

S

Φ(b, s, a) (p, Sb,s,a ψ)dμ(b, s, a).

Therefore, we obtain  (f, Sψt Φ) = (g (α) + p, Sψt Φ) =

S

Φ(b, s, a) (g (α) + p, Sb,s,a ψ)dμ(b, s, a)

 =

S

Φ(b, s, a) (f, Sb,s,a ψ)dμ(b, s, a),  

which concludes the proof.

Acknowledgments F. Bartolucci is part of the Computational Harmonic Analysis & Machine Learning unit of the Machine Learning Genoa Center (MalGa). S. Pilipovi´c and N. Teofanov were supported by MPNTR project no 174024, MNRVOID project no 19.032/961-103/19 and bilateral project ANACRES.

Appendix As mentioned in the introduction, both the ridgelet transform and the shearlet transform are related to the wavelet and the Radon transforms and, as we now show, they are mutually related as well. The Appendix is devoted to prove this connection, which has in part inspired our work. We start briefly recalling the ridgelet transform and we refer to [3] as a classical reference. We fix ψ ∈ S (R) and we define for every (θ, b, a) ∈ [−π, π ) × R × R+ the function Rθ,b,a ψ as 1 Rθ,b,a ψ(x) = ψ a



 x · n(θ ) − b , a

x ∈ R2 ,

where n(θ ) = (cos θ, sin θ ). Then, the ridgelet transform of f ∈ L1 (R2 ) with respect to ψ is given by  Rψ f (θ, b, a) =

R2

f (x)Rθ,b,a ψ(x) dx,

(θ, b, a) ∈ [−π, π ) × R × R+ .

The Shearlet Transform and Lizorkin Spaces

61

The ridgelet transform is related to the wavelet transform and the polar Radon transform by the following formula Rψ f (θ, b, a) = Wψ (R pol f (θ, ·))(b, a),

(19)

for every (θ, b, a) ∈ [−π, π ) × R × R+ and where the wavelet transform is onedimensional and acts on the variable q. By Eqs. (19) and (6), we obtain a relation formula between the ridgelet and the shearlet transform. We consider an admissible vectors ψ of the form (11) satisfying conditions (12) and with χ1 defined by (13) belonging to S (R). Proposition 3 For any f ∈ L1 (R2 ) ∩ L2 (R2 ) and ((b1 , b2 ), s, a) ∈ R2 × R × R+ , Sψ f ((b1 , b2 ), s, a)      v−s 1 b1 + vb2 a − 43 = |a| dv. Rχ1 f arctan v, √ ,√ φ2 √ 4 |a|1/2 1 + v2 1 + v2 1 + v2 R Proof By Theorem 1, for any f ∈ L1 (R2 ) ∩ L2 (R2 ) and ((b1 , b2 ), s, a) ∈ R2 × R × R× , the shearlet transform has the following expression 3

Sψ f ((b1 , b2 ), s, a) = |a|− 4



 R

Wχ1 (R aff f (v, ·))(b1 + vb2 , a)φ2

v−s |a|1/2

 dv. (20)

By Eq. (6), for every v ∈ R, (b1 , b2 ) ∈ R2 and a ∈ R+ , we compute Wχ1 (R aff f (v, ·))(b1 + vb2 , a) =√

1 1 + v2

= √ 4

1 1 + v2

· Wχ1 (R pol f (arctan v, √ ))(b1 + vb2 , a) 1 + v2 Wχ1 (D√

1+v 2

R pol f (arctan v, ·))(b1 + vb2 , a) 

b1 + vb2 a ,√ √ 1 + v2 1 + v2 1 + v2   b1 + vb2 a 1 . Rχ1 f arctan v, √ ,√ = √ 4 1 + v2 1 + v2 1 + v2

= √ 4

1



Wχ1 (R pol f (arctan v, ·))

Replacing formula (21) in (20) we obtain the desired relation.

(21)  

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References 1. F. Bartolucci, Radon transforms: Unitarization, Inversion and Wavefront sets [Ph.D. thesis]. http://hdl.handle.net/11567/997903. 2. F. Bartolucci, F. De Mari, E. De Vito, F. Odone. The Radon transform intertwines wavelets and shearlets. Applied and Computational Harmonic Analysis 47 (2019), no. 3, 822–847. 3. E. J. Candès, D. L. Donoho. Ridgelets: A key to higher-dimensional intermittency? Philos. Trans. R. Soc. A 357 (1999), no. 1760, 2495–2509. 4. S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H. Stark, and G. Teschke, The uncertainty principle associated with the continuous shearlet transform, International Journal of Wavelets, Multiresolution and Information Processing 6 (2008), no. 2, 157–181. 5. S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke, Shearlet coorbit spaces and associated Banach spaces, Applied and Computational Harmonic Analysis 27 (2009), no. 2, 195–214. 6. H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, Journal of Functional Analysis, 86 (1989), no. 2, 307–340. 7. H.G. Feichtinger, K. Gröchenig, Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions. Part II, Monatshefte für Mathematik, 108 (1989), no. 2–3, 129–148. 8. T. Fink, U. Kahler. A Space-Based Method for the Generation of a Schwartz Function with Infinitely Many Vanishing Moments of Higher Order with Applications in Image Processing, Complex Analysis and Operator Theory 13 (2019), no. 3, 985–1010. 9. G. B. Folland, A course in abstract harmonic analysis, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2nd ed., 2016. 10. P. Grohs, Continuous shearlet frames and resolution of the wavefront set, Monatshefte für Mathematik 164 (2011), no. 4, 393–426. 11. S. Helgason, The Radon transform, vol. 5 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 2nd ed., 1999. 12. A. Hertle. Continuity of the Radon transform and its inverse on Euclidean spaces. Math Z. 184 (1983), 165–192. 13. M. Holschneider. Wavelets. An analysis tool. The Clarendon Press, Oxford University Press, New York (1995). 14. L. Hörmander. The analysis of linear partial differential operators. I. Grundlehren der Mathematischen Wissenschaften 256, Springer-Verlag, Berlin (1983). 15. S. Jaffard. Exposants de Hölder en des points donnés et coefficients d’ondelettes. Comptes rendus de l’Académie des Sciences Series I Mathematics 308 (1989), no. 4, 79–81. 16. G. Kutyniok, D. Labate. Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc. 361 (2009), no. 5, 2719–2754. 17. G. Kutyniok, D. Labate. Shearlets. Appl. Numer. Harmon. Anal. Birkhäuser/Springer, New York (2012). 18. S. Mallat. A wavelet tour of signal processing, The sparse way. Elsevier, Academic Press, Amsterdam (2009). 19. S. Kostadinova, S. Pilipovi´c, K. Saneva, J. Vindas. The ridgelet transform of distributions. Integral Transforms Spec. Funct. 25 (2014), no. 5, 344–358. 20. S. Pilipovi´c, M. Vuleti´c. Characterization of wave front sets by wavelet transforms. Tohoku Math. J. 58 (2006), no. 3, 369–391. 21. S. Pilipovi´c, D. Raki´c, N. Teofanov, J. Vindas. The wavelet transforms in Gelfand-Shilov spaces, Collectanea Mathematica 67 (2016), no. 3, 443–460. 22. L. Schwartz. Théorie des distributions. Tome II. Actualités Sci. Ind., no. 1122 Publ. Inst. Math. Univ. Strasbourg 10. Hermann & Cie., Paris (1951). 23. S. Sonoda, N. Murata. Neural network with unbounded activation functions is universal approximator. Applied and Computational Harmonic Analysis 43 (2017), no. 2, 233–268. 24. F. Trèves. Topological vector spaces, distributions and kernels. Academic Press, New YorkLondon (1967).

Time–Frequency Localization Operators: State of the Art Federico Bastianoni

Abstract We present localization operators via the short-time Fourier transform. For both modulation and ultra-modulation spaces framework, well-known results about boundedness and Schatten-von Neumann class are reported. Asymptotic eigenvalues’ distribution and decay and smoothness properties for L2 eigenfunctions are exhibited. Eventually, we make a conjecture about smoothness of L2 -eigenfunctions for localization operators with Gelfand–Shilov windows and symbols in ultra-modulation spaces. Keywords Localization operators · Time–frequency analysis · Compact operators · Modulation spaces

1 Introduction ϕ ,ϕ

Localization operators are pseudodifferential operators Aa 1 2 introduced in 1988 by I. Daubechies [11]. They are a mathematical tool to localize a signal simultaneously in time and in frequency. A particular subset of this operators was already known since 1971 due to Berezin [4] under the name of anti-Wick operators. In this last framework, they arose as the result of a quantization rule in quantum mechanics, namely the anti-Wick quantization. Moreover, they had already been used as an approximation of pseudodifferential operators (“wave packets”) by Córdoba and Fefferman [10]. In this survey we shall present localization operators with a time–frequency analysis approach. For a different point of view, namely localization operators on groups, we address the reader to [33] by M. W. Wong.

F. Bastianoni () Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Torino, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_4

63

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F. Bastianoni

We will denote the Schwartz class by S and the space of tempered distributions by S  , ·,· will stand for the sesquilinear duality between S  and S which extends the usual inner product on L2 . For x, ω ∈ Rd , translation and modulation operators are defined as Tx f (t) := f (t − x) and Mω f (t) := e2π iω·t f (t), respectively. The short-time Fourier transform (STFT) of a signal f ∈ S  (Rd ) with respect to ϕ ∈ S (Rd )  {0} is the function on R2d defined by  Vϕ f (x, ω) := f, Mω Tx ϕ = f (t)e−2π iω·t ϕ(t − x) dt, (1) Rd

where the integral has to be understood in a weak sense. We suggest [20] for further details on the STFT. Given two non-zero windows ϕ1 , ϕ2 ∈ S (Rd )  {0}, analysis and synthesis window, respectively, and a symbol on the phase space a ∈ S  (R2d ), the ϕ ,ϕ localization operator Aa 1 2 : S (Rd ) → S  (Rd ) is defined by the (formal) integral  Aϕa 1 ,ϕ2 f (t) := a(x, ω)Vϕ1 f (x, ω)Mω Tx ϕ2 (t) dxdω, (2) R2d

equivalently we can give the weak definition as follows: Aϕa 1 ,ϕ2 f, g := a, Vϕ1 f Vϕ2 g ,

f, g ∈ S (Rd ).

(3)

ϕ ,ϕ

From (3) it comes immediately that Aa 1 2 is well defined and linear. Moreover, ϕ ,ϕ Aa 1 2 is continuous from S into S  endowed with the weak-∗ topology. Writing ϕ ,ϕ Aa 1 2 as in (2) makes clear the reason for the name localization operators. Think of f as a signal, that is, a suitable L2 -element, then we analyse the signal via its STFT. For the sake of simplicity consider the symbol a of type χΩ , where Ω ⊂ R2d is compact and χΩ is its characteristic function. The product aVϕ1 f is then the restriction of the analysed signal to the compact subset Ω in the phase ϕ ,ϕ space. Eventually we obtain the modified signal Aa 1 2 f multiplying by Mω Tx ϕ2 and integrating. Given a symbol σ ∈ S  (R2d ) the corresponding Weyl operator Lσ : S (Rd ) →  S (Rd ) is defined by Lσ f, g := σ, W (g, f ) ,

f, g ∈ S (Rd ),

(4)

where W (g, f ) is the cross-Wigner distribution of g and f :  W (g, f )(x, ω) :=

Rd

    t t −2π iω·t e g x+ f x− dt. 2 2

(5)

Lσ is well defined, linear and continuous. From the Schwartz Kernel Theorem [20, ϕ ,ϕ Theorem 14.3.4] there exists a unique symbol σ ∈ S  (R2d ) such that Aa 1 2 = Lσ . ϕ1 ,ϕ2 From a calculation in [5, 16, 25], we have Aa = La∗W (ϕ2 ,ϕ1 ) , hence the Weyl ϕ ,ϕ symbol of a localization operator Aa 1 2 is

Time–Frequency Localization Operators: State of the Art

65

σ = a ∗ W (ϕ2 , ϕ1 ).

(6)

The realization of a localization operator as a Weyl one is a technical tool used in many works, e.g. [3, 7, 9, 14, 15, 28], to deduce boundedness or compactness ϕ ,ϕ of Aa 1 2 according to the regularity of the symbol σ . Roughly speaking, the smoothing property of the convolution in (6) allows to consider even rough symbols a as long as we take sufficiently smooth windows ϕ1 , ϕ2 . In Sect. 2 we briefly recall modulation spaces, then present the main results about boundedness, compactness and Schatten-von Neumann class properties for ϕ ,ϕ operators Aa 1 2 . Section 3 is devoted to the study of eigenvalues and eigenfunctions of (compact) localization operators. In Sect. 4 Gelfand–Shilov spaces S (1) are introduced, boundedness and Schatten class properties for localization operators with ultra-distributional symbols and windows in ultra-modulation spaces are exhibited. Eventually, according to the result obtained in [3], we conjecture regularity ϕ ,ϕ properties for L2 -eigenfunctions of Aa 1 2 settled in this last framework. Throughout the paper A  B will mean that there exists a constant c > 0, independent of the parameters A and B may depend on, such that A ≤ cB.

2 Boundedness, Compactness and Schatten-von Neumann Class In this section we present the optimal class of windows and symbols for localization p,q operators, namely the modulation spaces Mm , then we report the fundamental ϕ ,ϕ result [8, Theorem 1] and the characterization for compact operators Aa 1 2 [14, Theorem 3.15]. The word optimal is used because of the fine properties that can be proved by making this choice, as shown later on. Recall that the Schatten-von Neumann class, Schatten class for short, Sp (L2 (Rd )) with 1 ≤ p < ∞ is the set of all compact linear√operators T on L2 such that the eigenvalues of the positive self-adjoint operator T ∗ T are in p . For consistency S∞ (L2 (Rd )) := B(L2 (Rd )), the set of all linear and bounded operators on L2 . Notice that Sp√⊆ Sq , 1 ≤ p ≤ q ≤ ∞. The norm of T in Sp is the p -norm of the eigenvalues of T ∗ T , for 1 ≤ p < ∞.

2.1 Modulation Spaces p,q

The right setting for localization operators are modulation spaces Mm introduced by H. G. Feichtinger in early 1980s for 1 ≤ p, q ≤ ∞, Banach case, [12]. The quasi-Banach version, 0 < p, q < 1, was studied in 2004 by Y. V. Galperin and S. Samarah, [17].

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A weight function will be a continuous, non-negative, even function. In the sequel we shall always denote by v a submultiplicative weight, i.e. v(z1 + z2 ) ≤ v(z1 )v(z2 ), and define the set Mv as the collection of all weight functions m which are vmoderate: m(z1 + z2 )  v(z1 )m(z2 ). In order to remain within the setup of Schwartz functions and tempered distributions, in this and next section we will consider only weights which have at most polynomial growth at infinity. We shall restrict our attention to the following weights: vs (z) := (1 + |z|2 )s/2 ,

vs ⊗ vr (x, ω) :=vs (x)vr (ω),

vs ⊗ 1(x, ω) := vs (x),

1 ⊗ vs (x, ω) :=vs (ω),

for s, r ∈ R. Consider 0 < p, q ≤ ∞, a fixed window ϕ ∈ S (Rd ){0} and a weight function p,q m ∈ Mv (R2d ). The modulation space Mm (Rd ) consists of all f ∈ S  (Rd ) such p,q p p,q 2d that Vϕ f ∈ Lm (R ). We write Mm if p = q and M p,q if m ≡ 1. The space Mm is (quasi-)Banach if endowed with the obvious (quasi-)norm "

f Mmp,q := Vϕ f Lp,q = m



Rd

# q1

q Rd

|Vϕ f (x, ω)|p m(x, ω)p dx

p

dω

when 0 < p, q < ∞, with natural modifications for p = ∞ or q = ∞. Different non-zero window functions yield equivalent (quasi-)norms, hence the same modulation space [17, 20]. Actually, as shown in [20, Theorem 11.3.7], in the previous definition the space of admissible windows S can be enlarged to Mv1 , the Feichtinger’s algebra. Also in this later class, different non-zero windows p,q generate the same space Mm . We state inclusion relations between modulation spaces, see for example [31, Proposition 1.2 (2)] and [17, Theorem 3.4]. Consider 0 < p1 ≤ p2 ≤ ∞, 0 < q1 ≤ q2 ≤ ∞ and m1 , m2 ∈ Mv (R2d ) such that m2  m1 . Then: p ,q1

Mm11

p ,q2

(Rd ) ⊆ Mm22

(Rd ).

(7)

From (7) and [20, Proposition 11.3.1 (d)] we get the following equality: S (Rd ) =

$ s≥0

where 0 < p, q ≤ ∞.

Mvp,q (Rd ), s

(8)

Time–Frequency Localization Operators: State of the Art

67

We briefly recall important convolution relations between modulation spaces. They were first proved in [7, Proposition 2.4] and extended to the quasi-Banach case ϕ ,ϕ in [3, Proposition 3.1]. Because of the form (6) of the Weyl symbol σ of Aa 1 2 , this relations are the essential tool to determine to which modulation space σ belongs to. In particular, this implies that we can take not only Schwartz functions as windows, but tempered distributions also. Proposition 1 Let w(ω) > 0 be an arbitrary weight function on Rd , and 0 < p, q, r, t, u, γ ≤ ∞, with 1 1 1 + = , u t γ and 1 1 1 + =1+ , p q r

for 1 ≤ r ≤ ∞

whereas p = q = r,

for 0 < r < 1.

For m ∈ Mv (R2d ), m1 (x) := m(x, 0) and m2 (ω) := m(0, ω) are the restrictions to Rd × {0} and {0} × Rd , and likewise for v. Then p,u

q,t d −1 (R ) 1 ⊗v2 w

Mm1 ⊗w (Rd ) ∗ Mv

r,γ

#→ Mm (Rd ).

(9)

By knowing in which modulation space σ is, it is possible to deduce properties for the localization operator written in the Weyl form. In this perspective, we report when we are able to extend a Weyl operator Lσ , defined from S into S  , to modulation spaces. Part (a) of the following result about Weyl operators Lσ was proved by J. Toft in [29, Theorem 4.3] and [30, Theorem 3.1], part (b) was done in [3, Theorem 3.3]. Theorem 1 (a) Consider 0 < p, q, γ ≤ ∞ such that 1/p + 1/q = 1/γ . If σ ∈ M p,min{1,γ } (R2d ), then Lσ : S (Rd ) → S  (Rd ) extends uniquely to a bounded linear operator from M q (Rd ) to M γ (Rd ). (b) If s, r ≥ 0, t ≥ r + s, and the symbol σ ∈ Mv∞,1 (R2d ), then Lσ : S (Rd ) → s ⊗vt S  (Rd ) extends uniquely to a bounded linear operator from Mv2r (Rd ) into Mv2r+s (Rd ). We shall denote by σP (T ) the point spectrum of T ∈ B(L2 (Rd )), i.e. the set of all the eigenvalues. We now exhibit [3, Proposition 3.6] on regularity for L2 -eigenfunctions of Weyl operators Lσ which will allow us to deduce easily Theorem 5.

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Proposition 2 Consider a Weyl symbol σ ∈ Mv∞,1 (R2d ) for some s > 0 and every s ⊗vt t > 0. If λ ∈ σP (Lσ )  {0}, then any eigenfunction f ∈ L2 (Rd ) with eigenvalue λ is in S (Rd ). (R2d ), for some s > 0 and Proof By Theorem 1, if the symbol σ is in Mv∞,1 s ⊗vt 2 d every t > 0, then Lσ acts continuously from L (R ) into Mv2s (Rd ). Starting now with the eigenfunction f in Mv2s (Rd ) and repeating the same argument with t ≥ s we obtain that the eigenfunction is in Mv22s (Rd ). Proceeding this way, using the inclusion relations for modulation spaces (7) and the equality (8) we infer f ∈

$

Mv2ns (Rd ) =

n∈N+

$

Mv2ns (Rd ) =

n∈N

$

Mv2s (Rd ) = S (Rd ).

s≥0

 

This concludes the proof.

Let us notice that, under the hypothesis of the previous proposition, Lσ is a compact operator on L2 . This follows from [5, Theorem 4.5], [21, Theorem 1.5 and Lemma 4.10] and the inclusion relations between modulations spaces (7). We shall remind this fact in the specific case of localization operators in the sequel.

2.2 Main Results on L2 ϕ ,ϕ

Sufficient conditions for Aa 1 2 to be in Sp have been obtained by E. Cordero and K. Gröchenig in [7]. From their subsequent paper, we report [8, Theorem 1] which includes the main result of the previous work and shows new necessary conditions ϕ ,ϕ for Aa 1 2 to be in the Schatten class Sp . Theorem 2 Let 1 ≤ p ≤ ∞. ϕ ,ϕ

(a) The mapping (a, ϕ1 , ϕ2 ) → Aa 1 2 is bounded from M p,∞ (R2d ) × M 1 (Rd ) × M 1 (Rd ) into Sp (L2 (Rd )) with a norm estimate

Aϕa 1 ,ϕ2 Sp ≤ B a M p,∞ ϕ1 M 1 ϕ2 M 1 ,

(10)

for a suitable B > 0. ϕ ,ϕ (b) Conversely, suppose that Aa 1 2 ∈ Sp (L2 (Rd )) for all windows ϕ1 , ϕ2 ∈ 1 d M (R ) and there exists a constant B > 0 depending only on the symbol a such that (10) holds true for every ϕ1 , ϕ2 ∈ S (Rd ). Then a ∈ M p,∞ (R2d ). It is natural to define the Schatten class Sp for 0 < p < 1 also, see [31]. The author expects to obtain an extension of the previous result in the case 0 < p < 1. In [14, Theorem 3.15], C. Fernández and A. Galbis gave a characterization of compact localization operators on L2 which states as follows.

Time–Frequency Localization Operators: State of the Art

69

Theorem 3 Let a ∈ M ∞ (R2d ) and g ∈ S (R2d ). Then the following are equivalent: ϕ ,ϕ

(a) Aa 1 2 is compact on L2 (Rd ) for every ϕ1 , ϕ2 ∈ S (Rd ); (b) For every R > 0 we have lim

sup |Vg a(x, ω)| = 0.

|x|→+∞ |ω|≤R

Let us observe that in the framework presented so far, M ∞ is the biggest symbol class admissible since M p,q (R2d ) ⊆ M ∞ (R2d ), for every 0 < p, q ≤ ∞. Conversely, S is the smallest window class admissible since S (Rd ) ⊂ M p,q (Rd ), for every 0 < p, q ≤ ∞. This fact is due to (6): the more we take rough symbols, the more we need smooth windows in order to have a fine Weyl symbol σ . For example, even an irregular symbol such as the Dirac’s delta distribution δ is allowed, since δ ∈ M 1,∞ (Rd ). This fact can be easily verified: Vϕ δ(x, ω) = δ, Mω Tx ϕ = e−2π iω·0 ϕ(0 − x) ∈ L1,∞ (R2d ) since ϕ(−·) ∈ S (Rd ) ⊂ L1 (Rd ).

3 Eigenvalues and Eigenfunctions The asymptotic behaviour of the eigenvalues’ distribution for localization operators with identical windows ϕ1 = ϕ = ϕ2 in L2 and symbol a = χΩ , where Ω ⊂ R2d is compact, has been object of interest. Under this assumption, it can be shown that ϕ,ϕ AχΩ is a compact and positive operator on L2 (Rd ) [5, 7, 22]. As an illustration we report a result as it was summarized in [1, Proposition 1.1]; it was proved with additional hypotheses on ∂Ω in [24] and in full generality in [13]. We shall denote by $V the cardinality of a set V . Given R > 0, R · Ω is the dilation of Ω ⊂ R2d and |R · Ω| its Lebesgue measure. Proposition 3 Consider ϕ ∈ L2 (Rd ) with ϕ 2 = 1 and let Ω ⊂ R2d be a compact set. Then for each δ ∈ (0, 1) ϕ,ϕ

${k : λk > 1 − δ, λk ∈ σP (AχR·Ω )} −→ 1, |R · Ω|

as R −→ +∞.

In [3], F. Bastianoni, E. Cordero and F. Nicola studied decay and smoothness ϕ ,ϕ properties for eigenfunctions in L2 of operators Aa 1 2 with Schwartz windows p,∞ 2d and symbol in M (R ), for some 0 < p < ∞, or Mv∞s ⊗1 (R2d ), for some s > 0. In both cases the corresponding localization operator is compact on L2 . If 1 ≤ p < ∞, then the operator is in Sp due to Theorem 2; if 0 < p < 1,

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then M p,∞ (R2d ) ⊂ M q,∞ (R2d ) for any 1 ≤ q ≤ ∞ and we apply the previous argument. As highlighted in [3, Remark 2.7], from [21] we have Mv∞s ⊗1 (R2d ) ⊂ M p,∞ (R2d ) ϕ ,ϕ2

Once again Aa 1

for any p > 2d/s.

is in Sp , therefore it is compact. It is easy to verify that (Aϕa 1 ,ϕ2 )∗ = Aa¯ 2

ϕ ,ϕ1

,

hence the self-adjointness property requires ϕ1 = ϕ2 and a real valued. Thus the localization operators considered in [3] are compact, but not necessarily self-adjoint. The next result, [3, Theorem 3.7], concerns the decay of L2 -eigenfunctions for ϕ1 ,ϕ2 Aa , as will be remarked below. Theorem 4 Consider a symbol a ∈ M p,∞ (R2d ), 0 < p < ∞, non-zero windows ϕ ,ϕ ϕ1 ,ϕ2 ϕ1 , ϕ2 ∈ S (Rd ) and assume that σP (Aa 1 2 )  {0} = ∅. If λ ∈ %σP (Aa γ )d {0}, 2 d any eigenfunction f ∈ L (R ) with eigenvalue λ satisfies f ∈ γ >0 M (R ). ϕ ,ϕ

The previous decay result implies that eigenfunctions of Aa 1 2 , with suitable symbol and windows, are extremely well-localized in the time–frequency space. This idea is formalized in [3] via Parseval % Gabor frames and description of Gabor coefficients’ decay. Notice that S  γ >0 M γ . The other main statement of the paper, [3, Theorem 3.10] about regularity, is the following. Theorem 5 Consider a symbol a ∈ Mv∞s ⊗1 (R2d ), for some s > 0, and non-zero ϕ ,ϕ ϕ ,ϕ windows ϕ1 , ϕ2 ∈ S (Rd ). Assume that σP (Aa 1 2 )  {0} = ∅. If λ ∈ σP (Aa 1 2 )  2 d {0}, any eigenfunction f ∈ L (R ) with eigenvalue λ belongs to the Schwartz class S (Rd ). Proof The assumption ϕ1 , ϕ2 ∈ S(Rd ) implies W(ϕ2 , ϕ1 ) ∈ S(R2d ) ⊂ (R2d ), for every r, t > 0. We next apply the convolution relations for Mv1,1 r ⊗vt ϕ ,ϕ modulation spaces (9), obtaining that Aa 1 2 = Lσ with σ ∈ Mv∞,1 (R2d ), for some s ⊗vt s > 0 and every t > 0. Hence the claim immediately follows by Proposition 2.  

4 Gelfand–Shilov Spaces Framework This section is devoted to develop a setting for localization operators which is slightly different from the one presented above. We shall consider a particular ϕ ,ϕ subspace of S and, by duality, obtain a class for windows and symbol for Aa 1 2  bigger than S . We report some boundedness and Schatten class results and, eventually, present a conjecture which will be object of a subsequent paper.

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71

4.1 Gelfand–Shilov Spaces The Gelfand–Shilov space S (1) (Rd ) := Σ11 (Rd ), see [18, 19], was defined through a projective limit for the first time by S. Pilipovi´c in [23]. For our aim, the following characterization will be sufficient [6, 26]. f ∈ S (Rd ) belongs to S (1) (Rd ) if one of the (equivalent) following conditions holds true: • supx∈Rd |x α f (x)|  h|α| α! and supω∈Rd |ωβ fˆ(ω)|  k |β| β! for every h, k > 0 and α, β ∈ Nd ; • supx∈Rd |x α f (x)|  h|α| α! and supx∈Rd |∂ β f (x)|  k |β| β! for every h, k > 0 and α, β ∈ Nd ; • supx∈Rd |f (x)|eh|x| < +∞ and supω∈Rd |fˆ(ω)|ek|ω| < +∞ for every h, k > 0;  where fˆ(ω) := Rd e−2π it·ω f (t) dt is the Fourier transform of f .  The strong dual S (1) (Rd ) is a space of tempered ultra-distributions and it contains S  (Rd ). As done for the tempered distributions, we can define the STFT  of f ∈ S (1) with respect to ϕ ∈ S (1) {0} via the sesquilinear duality ·,· defined  on S (1) × S (1) . For a fixed widow ϕ ∈ S (1) (Rd ) the following characterization holds: f ∈ S (1) (Rd ) ⇔ Vϕ f ∈ S (1) (R2d ).

(11)

For a proof of (11), and for a general view of Gelfand–Shilov type spaces in the time–frequency analysis spirit, we address the reader to [26, Theorem 3.8 and Theorem 3.9]. See [27, 32] also.

4.2 Ultra-Modulation Spaces We now drop the hypothesis on v and m to have at most polynomial growth at infinity. In particular, the only submultiplicativity implies that v is dominated by an exponential function, i.e. v(z)  ek|z| for some k > 0. In what follows we shall consider the exponential weights ws (z) := es|z| , 

for s > 0. Let us observe that ws is in S (1) , but does not belong to S  . Given m ∈ Mv (R2d ), 1 ≤ p, q ≤ ∞ and a non-zero window ϕ ∈ S (1) (Rd ),  p,q the ultra-modulation space Mm (Rd ) is the set of all f ∈ S (1) (Rd ) such that p,q p Vϕ f ∈ Lm (R2d ). Again, we write Mm if p = q and M p,q if m ≡ 1. Similarly to p,q the case of weights of at most polynomial growth at infinity, Mm are Banach spaces if endowed with the norms Vϕ · Lp,q ; the space of admissible windows S (1) can be m 1 enlarged to Mv and different windows yield equivalent norms. As done before, we can repeat the previous construction for 0 < p, q < 1 and get quasi-Banach spaces, e.g. [31].

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We recall the following characterizations for 0 < p, q ≤ ∞: S (1) =

$

Mwp,q , s



S (1) =

s≥0

&

p,q

M1/ws .

s≥0

4.3 Boundedness and Schatten Class 

Consider an ultra-distributional symbol on the phase space a ∈ S (1) (R2d ) and  ϕ ,ϕ windows ϕ1 , ϕ2 ∈ S (1) (Rd ). Then the localization operator Aa 1 2 : S (1) → S (1)  defined by (3), where ·,· is defined on S (1) × S (1) and f, g ∈ S (1) , is well defined, linear and continuous. This comes straightforward from (11) and the fact that S (1) is closed under pointwise multiplication. Even in the ultra-distributional framework we can define the Weyl transform  Lσ of a symbol σ ∈ S (1) (R2d ): Lσ f, g := σ, W (g, f ) for f, g ∈ S (1) . ϕ ,ϕ Analogously to the distributional case, Aa 1 2 can be realized as a Weyl operator with Weyl symbol σ = a ∗ W (ϕ2 , ϕ2 ), for a proof see, e.g., [28, Lemma 3.3]. We suggest [27] also for a further treatment of localization operators in this setting. The following result is due to E. Cordero, S. Pilipovi´c, L. Rodino and N. Teofanov, [9, Theorem 3.3]. It contains several sufficient conditions for boundedness on ultra-modulation spaces and Schatten class property. Theorem 6 ∞ (a) Consider s ≥ 0, a ∈ M1/1⊗w (R2d ) and ϕ1 , ϕ2 ∈ Mw1 s (Rd ). Then Aa 1 2 is s p,q d bounded on M (R ) for all 1 ≤ p, q ≤ ∞, moreover the operator norm satisfies the following uniform estimate: ϕ ,ϕ

∞

ϕ1 Mw1 ϕ2 Mw1 .

Aϕa 1 ,ϕ2 Op  a M1/1⊗w s

s

s

ϕ ,ϕ

(b) Consider 1 ≤ p ≤ 2. Then the mapping (a, ϕ1 , ϕ2 ) → Aa 1 2 is bounded from p,∞ p M1/1⊗ws (R2d ) × Mw1 s (Rd ) × Mws (Rd ) into Sp (L2 (Rd )), i.e.

Aϕa 1 ,ϕ2 Sp  a M p,∞ ϕ1 Mw1 ϕ2 Mwp . 1/1⊗ws

s

s

ϕ ,ϕ

(c) Consider 2 ≤ p ≤ ∞. Then the mapping (a, ϕ1 , ϕ2 ) → Aa 1 2 is bounded p,∞ p from M1/1⊗ws (R2d ) × Mw1 s (Rd ) × Mws (Rd ) into Sp (L2 (Rd )), i.e.

Aϕa 1 ,ϕ2 Sp  a M p,∞ ϕ1 Mw1 ϕ2

1/1⊗ws

s

p

Mws

.

It seems natural to the author expecting analogues results for the quasi-Banach case 0 < p < 1.

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We end this subsection remarking that boundedness properties for localization operators between ultra-modulation spaces were found by J. Toft in [31, Theorem 3.1].

4.4 Conjecture We conclude this survey with a conjecture in the spirit of Theorem 5, that will be object of a future work, see [2]. p,∞ Consider a symbol a ∈ M1/1⊗ws (R2d ), for some s > 0 and 1 ≤ p < ∞, ϕ ,ϕ and non-zero windows ϕ1 , ϕ2 ∈ S (1) (Rd ). Assume that σP (Aa 1 2 )  {0} = ∅. If ϕ1 ,ϕ2 λ ∈ σP (Aa )  {0}, any eigenfunction f ∈ L2 (Rd ) with eigenvalue λ belongs to S (1) (Rd ). Replacing S (1) with S (s) , s > 1/2, and considering the quasi-Banach setting 0 < p < 1 shall consist of the natural generalization for further research studies. Acknowledgments I express my gratitude to Professor Elena Cordero for her time and valuable advice.

References 1. L. D. Abreu, K. Gröchenig, and J. L. Romero. On accumulated spectrograms. Trans. Amer. Math. Soc., 368(5):3629–3649, 2016. 2. F. Bastianoni and N. Teofanov. Subexponential decay and regularity estimates for eigenfunctions of localization operators, 2020. J. Pseudo-Differ. Op. and Appl., to appear. ArXiv:2004.12947v2. 3. F. Bastianoni, E. Cordero and F. Nicola. Decay and Smoothness for Eigenfunctions of Localization Operators. J. Math. Anal. Appl., 492(2): 124480, 2020. https://doi.org/10.1016/ j.jmaa.2020.124480. 4. F. A. Berezin. Wick and anti-Wick symbols of operators. Mat. Sb. (N.S.), 86(128):578–610, 1971. 5. P. Boggiatto, E. Cordero, and K. Gröchenig. Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equations Operator Theory, 48(4):427–442, 2004. 6. J. Chung, S.-Y. Chung and D. Kim. Characterizations of the Gelfand-Shilov Spaces Via Fourier Transform. Proc. Amer. Math. Soc., 124(7):2101–2108, 1996. 7. E. Cordero and K. Gröchenig. Time-frequency analysis of localization operators. J. Funct. Anal., 205(1):107–131, 2003. 8. E. Cordero and K. Gröchenig. Necessary conditions for Schatten class localization operators. Proc. Amer. Math. Soc., 133(12):3573–3579, 2005. 9. E. Cordero, S. Pilipovi´c, L. Rodino, and N. Teofanov. Localization operators and exponential weights for modulation spaces. Mediterr. J. Math., 2(4):381–394, 2005. 10. A. Córdoba and C. Fefferman. Wave packets and Fourier integral operators. Comm. Partial Differential Equations, 3(11):979–1005, 1978. 11. I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34(4):605–612, 1988.

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12. H. G. Feichtinger. Modulation spaces on locally compact abelian groups. In Technical rep., University of Vienna, 1983, and also in “Wavelets and Their Applications”, pages 99–140. M. Krishna, R. Radha, S. Thangavelu, editors, Allied Publishers, 2003. 13. H. G. Feichtinger. and K. Nowak. A Szegö-type theorem for Gabor-Toeplitz localization operators. Michigan Math. J. 9, no. 1, 13–21, 2001. 14. C. Fernández and A. Galbis. Compactness of time-frequency localization operators on L2 (Rd ). J. Funct. Anal., 233(2):335–350, 2006. 15. C. Fernández and A. Galbis. Some remarks on compact Weyl operators. Integral Transforms Spec. Funct., 18(7–8):599–607, 2007. 16. G. B. Folland. Harmonic Analysis in Phase space. Princet. University Press, Princeton, NJ, 1989. p,q 17. Y. V. Galperin and S. Samarah. Time-frequency analysis on modulation spaces Mm , 0 < p, q ≤ ∞. Appl. Comput. Harmon. Anal., 16(1):1–18, 2004. 18. I. M. Gelfand and G. E. Shilov. Generalized Functions Vol. 3: Theory of differential equations. Translated from the Russian by Meinhard E. Mayer. Academic Press, New York-London, 1967. 19. I. M. Gelfand and G. E. Shilov. Generalized Functions Vol. 2. Academic Press, New YorkLondon, 1968. 20. K. Gröchenig. Foundations of time-frequency analysis. Appl. and Num. Harmon. Anal., Birkhäuser Boston, Inc., Boston, MA, 2001. 21. W. Guo, J. Chen, D. Fan and G. Zhao. Characterizations of Some Properties on Weighted Modulation and Wiener Amalgam Spaces Michigan Math. J., 68:451–482, 2019. 22. Jingde Du, M. W. Wong and Zhaohui Zhang. Trace class norm inequalities for localization operators. Integral Equations Operator Theory 41, no. 4, 497–503, DOI 10.1007/BF0122106. MR1857804 (2002f:47069), 2001. 23. S. Pilipovi´c. Tempered ultradistributions. Bol. Unione Mat. Ital., 7(2-B):235–251, 1988. 24. J. Ramanathan and P. Topiwala. Time-frequency localization and the spectrogram. Appl. Comp. Harmon. Anal. 1, no. 2, 209–215, 1994. 25. M. A. Shubin. Pseudodifferential operators and spectral theory. Springer-Verlag, Berlin, second edition, 2001. 26. N. Teofanov. Ultradistributions and Time-Frequency Analysis. Pseudo-Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 164. Birkhuser Basel, 2006. 27. N. Teofanov. Gelfand-Shilov spaces and localization operators. Funct. Anal. Approx. Comput., 7(2):135–158, 2015. 28. N. Teofanov. Continuity and Schatten–von Neumann properties for localization operators on modulation spaces. Mediterr. J. Math., 13(2):745–758, 2016. 29. J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal., 207(2):399–429, 2004. 30. J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Global Anal. Geom., 26(1):73–106, 2004. 31. J. Toft. Continuity and compactness for pseudo-differential operators with symbols in quasiBanach spaces or Hörmander classes. Anal. and Appl., 15(03):353–389, 2017. 32. J. Toft. Images of function and distribution spaces under the Bargmann transform. J. of PseudoDifferential Op. and App., 8 :83–139, 2017. 33. M. W. Wong. Wavelet transforms and localization operators. Operator Theory: Advances and Applications, volume 136, Birkhäuser Verlag, Basel, 2002.

Time-Frequency Analysis: What We Know and What We Don’t Leon Cohen

Abstract We discuss the basic ideas and motivations of time-frequency analysis from the point of view of what is known, what is not, what is mathematically possible, and what is not. We address the issue of whether a fully consistent theory of time-varying spectra is possible and the arguments given against the possible formulation of a proper theory. Historically, discussions of joint densities in time and frequency have been intimately tied with the question of the existence of manifestly positive joint densities, the marginals, and the uncertainty principle. We discuss the relations between these ideas. Keywords Time-frequency · Uncertainty principle · Positive distributions · Wigner distribution

1 Introduction The concept of spectral analysis as a fundamental tool for the investigation of matter originated with Bunsen and Kirchhoff and was immediately accepted because it was realized that the spectra are unique to each element. It rapidly became the main discovery tool for the nature of matter [14]. At that time, which was some 40 years before the acceptance of the idea that matter consisted of atoms, it was a total mystery as to why spectra should be unique to an element or compound. Of course, Fourier was the one that suggested that functions may be expanded in what we now call Fourier series, but his main aim was for solving the wave and heat equations. Indeed, it was Fourier who solved one of the major problems of his time, namely discovering the heat equation. Certainly, at that time, no one thought in terms of a spectrum changing in time. The first consideration of a changing spectrum was with the development of frequency modulation as a means of communication.

L. Cohen () Hunter College of The City University of New York, New York, NY, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_5

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Subsequently, it was realized that for many natural and man made signals, the spectra is clearly changing in time. While the theory of stationary spectral analysis is fully developed, at the present time there is no consistent theory or approach to describe spectra that changes in time. However, there have been many approaches suggested, and many reasons given as to why we may not be able to formulate a consistent mathematical and physical theory describing time-varying spectra. The aim of this paper is to examine the basic issues involved and the arguments that have been given as to why we cannot have a complete and consistent theory. The fundamental issues can be easily stated. There is universal agreement that if we have a time function, s(t), (a signal to some) then the energy in time is given by the absolute square of s(t) | s(t) |2 = intensity/energy per unit time at time t

(1)

and further, that the absolute value squared of the spectrum, S(ω), defined by 1 S(ω) = √ 2π



s(t)e−iωt dt

(2)

gives the intensity of frequencies | S(ω) |2 = intensity/energy per unit frequency at frequency ω

(3)

In the field of electrical engineering, | S(ω) |2 is called the energy density spectrum.

1.1 Why the Absolute Square of the Function? The energy density in time and in frequency each involve the absolute square of the respective function, s(t) and S(ω). Why the absolute square? Why not some other function? Why cannot the energy density be | s(t) |4 or any other functional of the signal? The reason we take Eqs. (1) and (3) is because nature has made that choice. They are predicted by the fundamental equations of physics and verified by experiment. In the electromagnetic case, for example, they are derived from Maxwell’s equations.

1.2 Why Do We Need a Time-Varying Spectrum Theory? The most fundamental reason is that we clearly have time-varying spectra in nature, and, if nature is doing it, we should be able to describe it. From a mathematical point of view one can readily convince oneself that there must be a joint description of time and frequency. Consider for example the time function

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77

s(t) = (α/π )1/4 e−αt

2 /2+iβt 2 /2

(4)

If one plots this function (say, its real part), then the number of oscillations, that is, the number of ups and downs, is clearly a function of time. But how do we describe that? There have been many approaches but none is totally satisfactory. Another example is the differential equation m

dv(t) + βv(t) = F (t) = white noise dt

(5)

which is the Brownian motion equation. What is the power spectrum of v(t)? The result was first obtained by Wang and Uhlenbeck [56]. It is given by | Sv (ω) |2 ∼

β2

1 + ω2

(6)

But what happened to time? Clearly the differential equation, Eq. (5), implies that we have a time dependent process. If we evolved to a steady state, how did we do so? Standard spectral analysis has no answer to these questions.

2 The Joint Density Approach Whenever we have two seemingly related densities and we believe they are related, then we should seek a joint density. Having the density in time or the density in frequency allows us to calculate averages of a time function or a frequency function, but does not allow us to calculate averages of functions of both time and frequency. That, of course, would be desirable because that would give us indication how the two variables, time and frequency are related and would allow the calculation of quantities such as correlations, conditional moments, instantaneous frequency, group delay, and other important physical quantities. In our case we have the densities |s(t)|2 and |S(ω)|2 , and hence it is natural to seek a joint density of time and frequency, P (t, ω), so that P (t, ω) = intensity/energy at time t and frequency ω The joint density should satisfy the marginals   P (t, ω) dt = | S(ω) |2 P (t, ω) dω = | s(t) |2 ;

(7)

(8)

Like any other density we want it to be manifestly positive P (t, ω) ≥ 0

(9)

In addition, we want P (t, ω) to give reasonable and “correct” answers for quantities that are joint in time and frequency.

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3 The Arguments Against a Proper Joint Density Theory Can one formulate a consistent approach that would yield a joint density P (t, ω) that satisfies the marginals and is manifestly positive? If not, why not? Over the years there have been many arguments made that a proper theory as just described is not possible. We review these arguments and ascertain their validity. We do not claim that we have a proper theory but just discuss the validity of arguments that have been made about the impossibility of having a proper theory.

3.1 Uncertainty Principle Argument Various phraseologies are commonly used, and statements like the following is typical: We cannot define a positive joint density because that would be a violation of the uncertainty principle. This statement is wrong as one can easily construct manifestly positive joint densities that satisfy the marginals and the uncertainty principle. The weak version of the uncertainty principle is σt2 σω2 ≥

1 4

(10)

where the standard deviations squared are defined by  σω2 =

(ω − ω )2 |S(ω)|2 d ω

(11)

(t − t )2 |s(t)|2 d t

(12)

 σt2 =

The proof of the uncertainty principle involves the consideration of  σt2 σω2

=

  (t − t ) |s(t)| d t 2

2

 (ω − ω ) |S(ω)| d ω 2

2

(13)

and moreover on the relationship between s(t) and S(ω) as given by Eq. (2). Therefore, it is clear that the uncertainty principle depends only on the marginals |s(t)|2 and |S(ω)|2 and on the fact that s(t) and S(ω) are Fourier transform pairs. Hence, any joint density which satisfies the marginals satisfies the uncertainty principle.

Time-Frequency Analysis

3.1.1

79

What Does the Uncertainty Principle Imply

• That there must be a relation between the marginals. They are functionally related by way of Eq. (2). The marginals cannot be chosen independent of each other. • It puts constraints on the possible types of joint distributions. Since both marginals cannot be too concentrated, the joint distribution cannot be too concentrated either. The fact that we cannot have joint distributions that are too concentrated does not mean that we cannot have joint distributions at all. • There is a stronger version of the uncertainty principle. If we write the signal in terms of amplitude and phase s(t) = A(t) eiϕ(t)

(14)

then σt2 σω2 ≥

1 4



1 + 4Cov2tω

 (15)

where Covtω =  t ϕ  (t) −  t  ω

(16)

with  t ϕ  (t) =



t ϕ  (t) | s(t) |2 dt

(17)

This is very revealing because it involves a quantity, Covtω , that appears to indicate that there is a relation between time and frequency. We emphasize that Eq. (15) follows only from the marginals.

3.1.2

Classical Examples of Joint Distributions with an Uncertainty Principle

To crystallize the issues of the uncertainty principle and the marginals, we construct classical type densities that have marginals so that an uncertainty principle exists. Consider a joint distribution of two variables, P (x, y), and for the moment consider the independent case P (x, y) = P (x)P (y).

(18)

Generally speaking the marginals are independent of each other and so are the standard deviations σy2 and σx2 , and therefore there is no uncertainty principle. But now take the two standard deviations to be

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σx =

' f 2 (η)

;

σy =

'

f 2 (η)(1 + g 2 (a))

(19)

where f (η) and g(a) are any two functions of the parameters η and a. We have that ' σx σy = f 2 (η) 1 + g 2 (a) (20) and therefore σx2 σy2 ≥ f 2 (η)

(21)

Thus, one sees that even for standard densities we can have an uncertainty principle if we make the marginals functionally dependent. To take a specific case, consider the joint density   1 1 x2 y2 ( (22) P (x, y) = exp − − 2πf 2 1 + g 2 2f 2 2f 2 (1 + g 2 ) This distribution is an independent joint distribution and is totally proper. The marginals are   x2 exp − 2 2f 2πf 2   1 y2 P (y) = ( exp − 2 2f (1 + g 2 ) 2πf 2 (1 + g 2 ) P (x) = (

1

(23)

(24)

No one would argue that there is anything fundamentally curious or that we have resolution problems or other metaphysical issues with Eq. (22). Moreover no one would argue that because we have an uncertainty principle, Eq. (21), a joint density does not exist. Also, we note that even though the marginals are related, the random variables, x and y, are not correlated. Notice further that if we make one marginal narrow we make the other broad. In addition we note that there are infinitely many joint densities with correlations that are consistent with the given marginals. For example, P (x, y) = ×

(

1

√ 1 + g2 1 − r 2 )  *+ 2 y 1 xy + exp − 2 x 2 − 2r ( 2f (1 − r 2 ) (1 + g 2 ) (1 + g 2 )

2πf 2

(25)

where r is any number between −1 and 1. For this case there is correlation, and the correlation is positive or negative, depending on the sign of r. Hence, as we have emphasized, from the marginals nothing can be concluded about the existence of a joint density or about the correlations between variables.

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3.2 The Impossibility Theorem of Wigner Numerous papers state or imply that Wigner has proven that no positive distributions exist that satisfy the marginals. The problem with this argument is that Wigner never said or proved that. What he did say in his 1932 paper is that there does not exist an expression for P which is bilinear and satisfies the marginals and is everywhere positive [57]. The word bilinear is crucial. Several years later, Wigner provided a proof [58]. Wigner was clear in both papers that he is assuming bilinearity. Bilinearity means that functionally the signal appears “twice” as a product. An important analysis of the Wigner proof and consequences has been given by MugurSchächter [45].

3.3 One Cannot Define Frequency at a Point in Time This is a pointless argument settled with the invention of the calculus. Analogous to this false argument is the argument that one cannot define velocity at a given time or point in space.

3.4 Quantum Mechanics Has Shown that One Cannot Measure Physical Quantities to Arbitrary Precision This is a total misunderstanding of quantum mechanics. There is nothing in quantum mechanics that implies we cannot measure a single quantity to arbitrary precision. Quite the contrary the frequency of spectral lines of atoms, for example, are explicitly calculated and have been experimentally verified.

3.5 Uncertainty Product Argument This argument is similar to the argument regarding the uncertainty principle but is usually stated in different terms: The uncertainty product can never be smaller than one-half and therefore we cannot measure the density within a smaller area. This statement totally confuses the meaning of standard deviations with the differential of the calculus that is used to define density of a continuous variable. To define a joint density, we would first obtain the value in a finite area ΔωΔt where Δ signifies an increment and take the limit as ΔωΔt → 0. But, the argument goes, that would be a violation of the uncertainty principle since ΔωΔt cannot approach zero. But it is not a correct argument. The uncertainty principle says σω σt ≥ 1/2 where the σ ’s are standard deviations. The confusion arises because sometimes Δ is used to denote standard deviation. The two usages of the notation “Δ” should not be confused.

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3.6 Time and Frequency Resolution “Trade Off” The phrase “trade off” between time and frequency resolution is often stated as the reason why a true time-frequency density is impossible. However this trade off is due to a particular method used to study the frequency properties of a function and is not inherent, as we now explain. An important method to study the frequency properties around time t is to modify the signal in such a way that it leaves the signal more or less unaltered around the time t and suppresses the signal for times far from t. This may be achieved by multiplying the time function by a window function, h(t) , to produce a modified signal, st (τ ), st (τ ) = s(τ )h(τ − t)

(26)

so that st (τ ) ∼

s(τ ) for τ near t . 0 for τ far from t

(27)

The short-time Fourier transform, St (ω), is then defined by the Fourier transform of the modified signal, st (τ ),  1 (28) St (ω) = √ s(τ )h(τ − t) e−iωτ dτ 2π and the time energy density by ! !2 ! 1 !! −iωτ |St (ω)| = s(τ )h(τ − t) e dτ !! ! 2π 2

(29)

A key aspect of this method is the selection of the window h(t). If the window is very long in time, little information about any time-varying spectral content will be discerned, and we are then just taking the spectrum of the signal. However for reasonably narrow windows, |St (ω)|2 estimates the frequency properties around time t. Now suppose we wish to study time properties at a particular frequency, ω. We window the spectrum of the signal, S(ω), with a frequency window function, H (ω), and define Sω (ω ) Sω (ω ) = S(ω ) H (ω − ω )

(30)

so that Sω (ω ) ∼

S(ω ) for ω near ω 0 for ω far from ω

(31)

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83

The short-frequency time transform is then  1  sω (t) = √ eiω t S(ω ) H (ω − ω )dω 2π

(32)

and the energy density is then ! !2 ! 1 !! iω t   ! |sω (t) | = e S(ω ) H (ω − ω )dω ! ! 2π 2

(33)

This estimates time properties around the frequency ω. Now, there is no inherent reason that h(t) and H (ω) have to be related. As an estimation problem, we can choose the appropriate h(t) to estimate some frequency property at time t and choose an unrelated H (ω) to estimate a time property at frequency ω. With this viewpoint, there are no issues with resolution trade off. However, if one insists, for whatever reason, on relating h(t) and H (ω) as Fourier transform pairs  1 H (ω) = √ (34) h(t) e−iωt dt 2π then we are forcing an uncertainty principle and we are forcing relations between St (ω) and sω (t). In such a case St (ω) = e−iωt sω (t)

(35)

and one can then define the joint time-frequency density, the spectrogram, by way of Psp (t, ω) = |sω (t) |2 = |St (ω)|2

(36)

In this case there is a trade off in some sense, since if we choose a very narrow time window, h(t), to estimate a frequency property at the time of interest, the corresponding spectral window, H (ω) , will be very broad and hence will give no information about the time properties at a given frequency. But note that it is we who have chosen this estimation technique. Windowing is not forced by nature or mathematics.

3.7 The Uncertainty Principle for the Spectrogram Since we have modified the signal one can ask for the uncertainty principle of the modified signal. The mean time and frequency for the spectrogram are 

 t =

t Psp (t, ω)dtdω

;

ω =

ω Psp (t, ω)dtdω

(37)

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and similarly   t2 = t 2 Psp (t, ω)dtdω

 ;

 ω2 =

ω2 Psp (t, ω)dtdω.

Defining the standard deviations in the usual way, one obtains that ! ! ! ! σt2 = σt2 ! + σt2 ! signal

window

(38)

(39)

and ! ! σω2 = σω2 !

signal

! ! + σω2 !

window

(40)

! σt2 !signal is the standard deviation for the signal and the other expressions similarly denoted. Using these expressions one obtains the uncertainty principle for the spectrogram [13, 21, 59] σω σt ≥ 1.

(41)

This uncertainty principle is different than the usual one where the right hand side is 1/2. Notice that, again, this uncertainty principle follows only from the marginals of the spectrogram. Thinking of the spectrogram as a density without any philosophical interpretation, it is certainly proper and has an uncertainty principle given by Eq. (41). Does the uncertainty principle for the spectrogram imply that there cannot exist a manifestly positive density satisfying the marginals of the spectrogram? (Eqs. (46) and (47) which are given in Sec. 4.2). Clearly not, since indeed it is derived using the spectrogram, Eq. (36), which is manifestly positive.

4 Classification of Time-Frequency Densities We now discuss the types of densities that have been developed and the relation to the satisfaction of the marginals and the question of positivity.

4.1 Bilinear Distributions Satisfying the Marginals All bilinear densities can be characterized by [9], 1 P (t, ω) = 4π 2



s ∗ (u −

−iθt−iτ ω+iθu   dθ 2 τ ) s(u + 2 τ )Φ(θ, τ )e

dτ du (42)

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85

where Φ(θ, τ ), called the kernel, is a function that characterizes the particular density [1, 6–8, 11, 12, 28–32, 34, 60]. For bilinearity Φ(θ, τ ) must not be a functional of s(t). To satisfy the marginals one must take Φ(θ, 0) = Φ(0, τ ) = 1.

(43)

Alternatively one can write  P (t, ω) =

K(t, ω; x  , x)s ∗ (x  )s(x)dx  dx

(44)

where now K(t, ω; x  , x) characterizes the joint density. For bilinear joint densities K must be independent of s(t). These bilinear forms cannot be manifestly positive and satisfy the marginals of s(t). That is what Wigner proved. If the kernel is taken to be a functional of s(t), then manifestly positive distributions that satisfy the marginals do exist [10].

4.2 Manifestly Positive Bilinear Densities Not Satisfying the Marginals These type of distributions go by many names depending on the field, most commonly they are called spectrograms. We have discussed them in Sec. 3.6. They are given by ! !2  ! 1 ! −iωτ ! P (t, ω) = ! √ s(τ ) h(τ − t)dτ !! e 2π

(45)

where h(t) is a window function. They do not satisfy the marginals of the signal, but have marginals that intermingle the signal and window  P (ω) =

 P (t, ω)dt =

 P (t) =

|S(ω )|2 |H (ω − ω)|2 dω

(46)

|s(τ )|2 |h(τ − t)|2 dτ.

(47)

 P (t, ω)dω =

The spectrogram may be obtained from the general class, Eq. (42), by taking the kernel to be  Φ(θ, τ ) = h∗ (u − 12 τ ) h(u + 12 τ ) e−iθu du. (48)

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L. Cohen

4.3 Non-bilinear, Manifestly Positive Distributions Satisfying the Marginals Consider the density given by P ( t, ω) = |S(ω) |2 |s( t) |2 .

(49)

This is a proper density function, satisfies the marginals, and from which the uncertainty principle follows. This joint density is an independent density since the joint is a product of the marginals. Of course the variables are in some sense related because the marginals are related as we discussed in Sec. 3.1. We do not claim that nature follows this density, but it explicitly shows that many of the arguments used to disprove existence of positive joint densities are false. To the best of the author’s knowledge, the first to use this density to clarify arguments on joint densities was Margenau [43]. There are many ways to generate manifestly positive distributions that satisfy the marginals and have correlations [10, 19, 20, 22, 27, 33, 35, 36, 36, 38, 39, 46–54]. One way is [10] P ( t, ω) = |S(ω) |2 |s( t) |2 Ω(u, v)

(50)

where Ω(u, v) is a function of u and v. For u and v we substitute  u( t) =

t −∞



|s( t  ) |2 d t 

;

v(ω) =

ω −∞

|S(ω ) |2 dω

(51)

To satisfy the marginals, the conditions on Ω are 

1

 Ω(u, v)dv =

0

1

Ω(u, v)du = 1.

(52)

0

To illustrate with an example consider the signal s(t) = (α/π )1/4 e−αt

2 /2+iβt 2 /2

(53)

with α and β real. The spectrum is , S(ω) =

√

√

α 2 e−ω /2(α−iβ) . π (α − iβ)

(54)

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87

The densities in time and frequency are given by | s(t) |2 = | S(ω) |2 =

α −αt 2 e π

(55)

α 2 2 2 e−αω /(α +β ) . 2 +β )

(56)

α2 2 2 2 2 e−αt −αω /(α +β ) 2 +β )

(57)

π(α 2

Consider first the density , P (t, ω) =

π 2 (α 2

this is clearly an independent density and a perfectly proper one. It satisfies the marginals but if one were to be given Eq. (57) and if the marginals were calculated, it would be noticed that the marginals are related because the parameter α appears in both marginals. To illustrate Eq. (50) we calculate u and v as per Eqs. (51) u( t) = v(ω) =

α π



t

e−αx d x 2

−∞

α 2 π(α + β 2 )



(58)

ω

e−αx

−∞

2 /(α 2 +β 2 )

dx.

(59)

Using the general formula √



1 2π σ 2

x

−∞

e

− (x−μ) 2 2σ

2

dx =

1 2



 1 + erf

x−μ √ σ 2

 (60)

where 2 erf(x) = √ π



x

e−x dx 2

(61)

0

we obtain that √ / 1. 1 + erf αt 2   1 α v(ω) = 1 + erf ω . 2 π(α 2 + β 2 ) u( t) =

(62) (63)

Now take Ω(u, v) = 2(u + v − 2uv) = 1 − (2u − 1)(2v − 1)

(64)

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L. Cohen

which satisfies Eq. (52). Evaluation of Ω(u, v) leads to  √  Ω(u, v) = 1 + erf αt erf

α ω 2 π(α + β 2 )

 (65)

and gives ,

α2 2 2 2 2 e−αt −αω /(α +β ) + β 2)   √  α 1 + erf ω αt erf π(α 2 + β 2 )

P (t, ω) =

π 2 (α 2

(66)

which is a proper density and satisfies the uncertainty principle.

5 Why Is There an Exception to Wigner’s Theorem? For the signal s( t) = (α/π )1/4 e−α t

2 /2+iβ t 2 /2

(67)

the Wigner distribution W (t, ω) =

1 2π



s ∗ (t − 12 τ ) s(t + 12 τ ) e−iτ ω dτ

(68)

is calculated to be W ( t, ω) =

1 −α t 2 −(ω−β t)2 /α e π

(69)

which is manifestly positive. Certainly the Wigner distribution as defined by Eq. (68) appears to be bilinear and yet it is manifestly positive for s( t) as given by Eq. (67). This appears to be contrary to Wigner’s theorem. Why should there be an exception? First we note that this shows that the issue of whether the density goes negative or not is not related to the uncertainty principle, even for the Wigner distribution. To understand why there is an exception to Wigner’s theorem we first consider the special case of Eq. (67) by taking β = 0 s( t) = (α/π )1/4 e−α t

2 /2

.

(70)

Time-Frequency Analysis

89

The energy densities are | s(t) | = 2

α −αt 2 e π

;

| S(ω) | = 2

1 −ω2 /α e πα

(71)

and the Wigner distribution is W ( t, ω) =

1 −α t 2 −ω2 /α e . π

(72)

We rewrite Eq. (72) as 1 −α t 2 −ω2 /α W ( t, ω) = = e π

-

α −αt 2 e π

-

1 −ω2 /α e πα

(73)

and hence W ( t, ω) = | s(t) |2 | S(ω) |2 .

(74)

Thus we see that even though the Wigner distribution is bilinear in general, for this case it also belongs to the non-bilinear class given by Eq. (50) with Ω = 1. Now consider the following density P ( t, ω) = | s(t) |2 | S(ω) |2 ρ(t, ω)

(75)

with -

α −αt 2 e π α 2 2 2 2 e−αx /(α +β ) | S(ω) | = π(α 2 + β 2 ) | s(t) | = 2

(76) (77)

and where   2 2 2 2 ρ(t, ω) = 1 + (β/α)2 e−(ω−βt) /α+αω /(α +β ) .

(78)

The joint density is manifestly positive and the marginals are given by Eqs. (76) and (77). Equation (75) which is manifestly positive is a proper joint density with correlations and no one would ever argue that there is anything wrong with it or that we have a resolution problem or a measurement problem or that there are mysterious aspects to it. Simplification of Eq. (75) yields Eq. (69). The reason that it is manifestly positive is that it is not bilinear in the signal.

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6 Is Time-Frequency “Hidden” in Standard Fourier Analysis? For a signal, s(t), and its associated spectrum, S(ω), the average frequency is  ω = ω | S(ω) |2 dω. (79) How can one bring time into this expression? Suppose we write the signal in terms of amplitude, A(t), and phase, ϕ(t), s(t) = A(t) eiϕ(t) in which case 1 S(ω) = √ 2π



(80)

A(t) eiϕ(t) e−iωt dt.

(81)

Substituting Eq. (81) into Eq. (79), one obtains  ω = ϕ  (t) A2 (t) dt.

(82)

One is tempted to define Instantaneous frequency = ωi (t) = ϕ  (t).

(83)

Similarly the square of the standard deviation in frequency, defined by  σω2 =

( ω −  ω )2 | S(ω) |2 dω

(84)

may be expressed as   σω2 =

A (t) A(t)

2

 A2 (t) dt +



ϕ  (t) −  ω

2

A2 (t) dt.

(85)

The interesting aspect of Eqs. (82) and (85) is that they express important frequency averages in terms of the time function and hence imply that they are somehow related [11, 12]. Similar expressions arise in standard probability densities. For a probability distribution of two variables, P (x, y), the average value of x and the conditional value of x for a fixed y are respectively   x =

xP (x, y)dxdy

(86)

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91

and 1  x y = P (y)

 (87)

xP (x, y)dx

where P (y) is the marginal of the y variable,  P (y) =

(88)

P (x, y)dx.

Also, the standard deviation of x and the conditional standard deviation are respectively given by  σx2 =

(x −  x )2 P (x, y)dxdy

(89)

and 2 σx|y

1 = P (y)

 (x −  x y )2 P (x, y)dx.

(90)

The relation between the two is    2 2 2  x y −  x P (y) dy σx = σx|y P (y) dy +

(91)

where P (y) is the marginal distribution of y. Comparing Eqs. (82) and (85) with Eqs. (87) and (91) one is tempted to take 

 ω t = ϕ (t)

 ;

2 σω|t

=

A (t) A(t)

2 .

(92)

What Eqs. (82) and (85) imply is that there are some kind of relations between time and frequency and that these relations are similar to expressions obtained in standard probability theory. The full meaning of what these relations imply is an unsolved problem [16, 17, 37].

7 Can One Derive Time-Frequency Densities? Historically, the first seemingly plausible derivation was given by Moyal [44] in the quantum mechanical case and the identical derivation was then given by Ville [55] for the time-frequency case. They obtained the Wigner distribution. Subsequently, Cohen [9] generalized Moyal’s approach to obtain the totality of bilinear forms, Eq. (42).

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L. Cohen

One defines the time and frequency operators by T and W where,  T =

 W =

t

in the time representation

d i dω

in the frequency (Fourier) representation

1 d i dt

in the time representation

ω

in the frequency representation.

(93)

(94)

The average of any time function, f (t), can be obtained either from the signal or its spectrum  f (t) =



S ∗ (ω) f

f (t) | s(t)|2 dt =

 −

 1 d S(ω) dω. i dω

(95)

For a function of ω, g(ω),  g(ω) =

 g(ω) |S(ω)| dω = 2

∗

s (t) g



1 d i dt

 s(t) dt.

(96)

Moyal argued that for the two variable case we generalize the above in the following way. In standard probability theory the characteristic function, M(θ, τ ) is defined as the average of eiθt+iτ ω ,  M(θ, τ ) = eiθt+iτ ω =

eiθt+iτ ω P (t, ω) dt dω.

(97)

The density function is given by P (t, ω) =

1 4π 2



M(θ, τ )e−iθt−iτ ω dθ dτ.

(98)

Moyal argued that we can find the average of eiθt+iτ ω by substituting for the ordinary variables, the time and frequency operators,  M(θ, τ ) =

s ∗ (t) eiθ T +iτ W s(t)dt.

(99)

s ∗ (t − 12 τ ) s(t + 12 τ )eiθt dt.

(100)

This straightforwardly leads to  M(θ, τ ) =

Time-Frequency Analysis

93

and when substituted into Eq. (98) results in P (t, ω) =

1 2π



s ∗ (t − 12 τ ) s(t + 12 τ ) e−iτ ω dτ

(101)

which is the Wigner distribution. Cohen argued that there 0 are many other operator 1 forms that one can take for 1 iθt+iτ ω iθ T iτ W iτ W iθ T , or eiθ T /2 eiτ W eiθ T /2 , and . For example, 2 e e +e e e showed that all cases may be parameterized by taking the characteristic function to be  M(θ, τ ) = Φ(θ, τ ) s ∗ (t) eiθ T +iτ W s(t) dt (102) which evaluates to  M(θ, τ ) = Φ(θ, τ ) s ∗ (t −

iθt  2 τ ) e s(t

+

 2 τ ) dt

(103)

Substituting this into Eq. (98) results in P (t, ω) =

1 4π 2



s ∗ (u − 2 τ )s(u + 2 τ )Φ(θ, τ ) e−iθt−iτ ω+iθu dθ dτ du (104)

which is the generalized distribution, Eq. (42).

7.1 A New Approach Recently, at the suggestion of Scully, a new approach has been presented for deriving joint densities [3, 5]. One starts with the expectation value of an operator, G, defined by  G =

s ∗ (t)G(T ,W )s(t)dt

(105)

and asks whether this expression can always be expressed in the following way: 2 an ordinary function of t and ω G = that depends only on s(t) 2 an ordinary function of t and ω × dt dω. that depends only on the operator G 

(106)

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L. Cohen

If this is possible, then we would conclude that  G =

(107)

P (t, ω)g(t, ω)dt dω

where the density depends only on the signal and is given by P (t, ω) =

an ordinary function of t and ω that depends only on s(t)

2 (108)

and that g(t, ω) depends only on the operator G and is given by g(t, ω) =

2 an ordinary function of t and ω . that depends only on the operator G

(109)

Equation (109) is called a correspondence rule since it relates an operator, G, with an ordinary function g(t, ω) . Expressing Eq. (105) as Eq. (106) is always possible but it is not unique, and the non-uniqueness generates all possible joint densities. In contract to previous derivations, in this approach both the joint density and corresponding association of operators with ordinary functions are obtained by the same derivation.

8 What Did Wigner Do in His 1932 Paper? It is often not appreciated what Wigner did in his famous 1932 paper [57]. Besides introducing the Wigner distribution, he introduced a new method to understand, solve, and approximate solutions to the Schrödinger equation. At that time, the Schrödinger equation was already applied to atoms and molecules, but the field of quantum statistical mechanics was not yet developed. Of particular importance is that Wigner calculated an approximation to the quantum correction to the second viral coefficient of a gas. Margenau and others had made a first attempt to understand the quantum corrections. Wigner’s approach is to use the classical statistical mechanical formula for the second virial coefficient but substituted the Wigner distribution for the classical distribution function. That sneaked in the quantum mechanics. The Schrödinger equation is a partial differential equation, and what Wigner did is to transform the spatial differential equation into a differential equation in phase space, defined by way of the Wigner distribution. It is important to appreciate that the Wigner distribution is not obtained by solving Schrödinger equation and then substituting the solution into the Winger distribution; what he did was obtain the differential equation for the Wigner distribution which is equivalent to the Schrödinger equation.

Time-Frequency Analysis

95

Galleani and Cohen [23–26] generalized the method to both ordinary and partial differential equations. They found that considerable mathematical and physical richness results. We discuss this in the next section.

8.1 The Wigner Distribution as a Tool for Solving Differential Equations Fourier solved one of the major problems of his time: He obtained the heat equation. This was sometime after D’Alembert and Euler obtained the wave equation [14]. The main motivation of Fourier in devising Fourier analysis was to find a method to solve the wave and heat equations. Of course, the method is also applicable to ordinary differential equations. In the Fourier method one transforms the differential equation from position space into wavenumber space. We know that in analyzing a function using the Wigner distribution, it is often the case that the understanding of the function is revealed in a much clearer way than in just position or just wavenumber space. Hence, we argue that transforming a position differential equation into phase space may reveal new insights into the nature of the original differential equation. That is, we want to transform linear ordinary and partial differential equations into phase space as Wigner did for the Schrödinger equation. Galleani and Cohen [23–26] developed the methods to do that and have found that (a) the dynamics are easier to understand, (b) solving the dynamic equations in phase space is often easier than in position or in momentum space,(c) that it leads considerable insight, and (d) that it leads to new approximation methods. We do not discuss the details as to how to transform differential equations into phase space equations, but we give some examples. For a time dependent position function, u(x, t), the Wigner distribution for position-wavenumber (xk space) at time t is  1 W (x, k, t) = (110) u∗ (x − 12 τ, t)u(x + 12 τ, t)e−iτ k dτ 2π We now give some examples to show that new insights are obtained when we transform standard equations into phase space equations. Consider the two famous partial differential equations ∂u(x, t) ∂ 2 u(x, t) = ia ∂t ∂x 2

(111)

∂ 2 v(x, t) ∂v(x, t) =D ∂t ∂x 2

(112)

The first being the Schrödinger free particle equation and the second the diffusion equation. There have been many attempted comparisons over the years and many authors argued that the way to compare these two equations is to make the diffusion

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L. Cohen

coefficient complex by taking D = ia. This is highly unsatisfactory because indeed the solutions are dramatically different. Consider the transformation to k space where 1 S(k, t) = √ 2π



u(x, t)e−ixk dk

(113)

Equivalent to Eqs. (111) and (112) we have ∂Sv (k, t) = −iak 2 Sv (k, t) ∂t ∂Su (k, t) = −Dk 2 Su (k, t) ∂t

(114) (115)

This is the standard Fourier analysis procedure. Still, the equations appear similar. However if we transform into phase space we obtain that [24] ∂Wu (x, k, t) ∂Wu = −2ak ∂t ∂x

(116)

∂Wv (x, k, t) D ∂ 2 Wv = − 2Dk 2 Wv ∂t 2 ∂x 2

(117)

Equations (116) and (117) are very different and therefore we would expect the solutions to be different both conceptually and mathematically as indeed they are. Another example is the Brownian motion equation dv(t) + βv(t) = F (t) = white noise dt

(118)

The statistical properties of F (t) are E[F (t)] = 0 E[F (t1 )F (t2 )] = N0 δ(t1 − t2 )

(119) (120)

Wang and Uhlenbeck showed that the power spectrum of v(t) is given by | Sv (ω) |2 ∼

1 β 2 + ω2

(121)

As mentioned in Sect. 1.2, time has somehow disappeared, but clearly the differential equation, Eq. (118), implies that we have a time dependent process. Now, the equation for the Wigner spectrum W v (t, ω), which is equivalent to Eq. (118) is [26] 1 ∂2 ∂ W v (t, ω) + β W v (t, ω) + (β 2 + ω2 )W v (t, ω) = W F (t, ω) 4 ∂t 2 ∂t

(122)

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97

where W F (t, ω) is the Wigner spectrum of white noise. The solution to Eq. (122) is 1 W v (t, ω) = π −

  1 N0 −2βt sin 2ωt N0 2 E[v0 ] − e + 2β ω 2π β 2 + ω2

N0 e−2βt (cos 2ωt − ω/β sin 2ωt) 2π β 2 + ω2

;t ≥ 0

(123)

At infinite time the surviving term is indeed the power spectrum of v(t) as given by Eq. (121). Hence the Wigner spectrum shows how the process evolves in time. To show the effectiveness of the method in obtaining approximate solutions, consider the case of wave propagation. One usually categorizes wave motion as standing waves (periodic waves) and moving waves. There is a long history to the theory of moving waves, and indeed many names have been applied to these types of waves. Among them are pulses, wave groups, transient waves, progressive waves, wave packets, nonrecurring waves, and traveling waves. Transforming the general wave equation N  n=0

∂ nu  ∂ nu = bn n ∂t n ∂x M

an

(124)

n=0

into phase space results in new insights and new approximations. The standard classical method is to substitute eikx−iωt into Eq. (124) to obtain an algebraic equation N 

an (iω)n =

n=0

M 

bn (ik)n

(125)

n=0

One then solves for ω as a function of k ω = ω(k)

(126)

which is called the dispersion relation. Generally there are multiple complex solutions and each is called a mode ω = ωR + iωI

(127)

The group velocity is the derivative of the real part of ω v(k) =

dωR (k) dk

(128)

Lossless dispersive propagation is when ω(k) is real and dispersive propagation with damping is when ω(k) is complex. Each mode satisfies the differential equation

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L. Cohen

  1 ∂ ∂u(x, t) = ω u(x, t) i ∂t i ∂x

(129)

If one transforms Eq. (129) into the Wigner domain the differential equation is i

     ∂ 1 ∂ 1 ∂ W (x, k, t) = ω k + − ω∗ k − W (x, k, t) ∂t 2i ∂x 2i ∂x

(130)

If we now expand the right hand side we obtain ∂ i W (x, k, t) = ∂t

)

(2)

2

∂ ∂ 2ωI (k) − vg (k) ∂x − 14 ωI (k) ∂x 2   1 (3) ∂ 3 + 24 ωR (k) ∂x + ···

+ W (x, k, t)

(131)

Keeping the first two terms we have   ∂ ∂ Wa (x, k, t)  2ωI (k) − vg (k) Wa (x, k, t) ∂t ∂x

(132)

which gives the approximation [2, 40, 41] Wa (x, k,t)  e2ωI t W (x − vg t, k, 0)

(133)

This approximation clearly shows that each phase space point propagates with constant velocity given by the group velocity at the phase space point. Furthermore, notice the simplicity of the approximation. To obtain Wa (x, k,t), no calculations are required, just a simple substitution in the Wigner distribution at the initial time. While the approximation given by Eq. (133) has been shown to be an excellent one, the approximate Wigner distribution is not a proper one in that it is not representable. That is, in general, there does not exist a wave function that generates it. The general issue of why some nonrepresentable Wigner functions nonetheless give excellent results is a fascinating and important subject [4, 15, 18, 42].

References 1. R. G. Baraniuk and D. L. Jones, IEEE Transactions on Signal Processing, 43, 2269–2282, 1995. 2. J. S. Ben-Benjamin and L. Cohen, “Pulse propagation and windowed wave functions,” Journal of Modern Optics, 61, 36–42, 2014. 3. J. S. Ben-Benjamin, M. B. Kim, W. P. Schleich, W. B. Case, L. Cohen, “Working in phase space with Wigner and Weyl,” Fortschr. Phys. 65 , 1–11, 2017. 4. J. S. Ben-Benjamin, L. Cohen , N. C. Dias, P. Loughlin, and J. N. Prata, “What is the Wigner function closest to a given square integrable function?”, Siam J. Math. Anal., 50, 5161–5197, 2018.

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Some Notes About Distribution Frame Multipliers Rosario Corso and Francesco Tschinke

Abstract Inspired by a recent work about distribution frames, the definition of multiplier operator is extended in the rigged Hilbert spaces setting and a study of its main properties is carried on. In particular, conditions for the density of domain and boundedness are given. The case of Riesz distribution bases is examined in order to develop a symbolic calculus. Keywords Distributions · Rigged Hilbert spaces · Frames · Multipliers

1 Introduction Bessel multipliers were introduced by Balazs in [8] and they became objects of several works [9, 40–45]. Multipliers have been studied also in particular cases [15, 21, 30] and found applications in physics, signal processing, acoustics, and mathematics. To define them we need to recall some notions (see [16, 31]). Let H be a Hilbert space with inner product ·|· and norm · . A sequence φ = {φn }n∈N is a Bessel sequence of H with upper bound B > 0 if 

|f |φn |2 ≤ B f 2 ,

∀f ∈ H .

n∈N

A sequence φ = {φn }n∈N is a (discrete) frame if there exist A, B > 0 such that A f 2 ≤



|f |φn |2 ≤ B f 2 ,

∀f ∈ H .

n∈N

R. Corso () · F. Tschinke Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_6

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Let φ = {φn }n∈N , ψ = {ψn }n∈N be two sequences of H and m : N → C. The operator Mm,φ,ψ defined by Mm,φ,ψ f =



mn f |ψn φn ,

n∈N

3 and with domain the subspace of f ∈ H such that n∈N mn f |ψn φn is convergent, is called the multiplier of φ, ψ with symbol m. When φ, ψ are Bessel sequences (frames) and m is a bounded sequence, then Mm,φ,ψ is defined on H , bounded, and it is called a Bessel (frame) multiplier. Independently in [1] and in [33], the notion of continuous frame was introduced as generalization of discrete frame and later in [10] the correspondent notion of Bessel continuous (frame) multiplier was formulated, which we now recall. The setting involves a measure space (X, μ) with positive measure μ. A map F : x ∈ X → Fx ∈ H is called a continuous frame with respect to (X, μ) if 1. F is weakly measurable, i.e. f → f |Fx is μ-measurable for every f ∈ H ; 2. there exist A, B > 0 such that  |f |Fx |2 dμ ≤ B f 2 , ∀f ∈ H . (1) A f 2 ≤ X

A weakly measurable map F : x ∈ X → Fx ∈ H is called a Bessel continuous map with respect to (X, μ) if the second inequality in (1) holds. Let F, G : X → H be Bessel continuous maps and m ∈ L∞ (X, μ). An operator Mm,F,G can be weakly defined by  Mm,F,G f |g =

m(x)f |Fx Gx |g dμ,

f, g ∈ H .

X

This operator is called the Bessel continuous multiplier of F, G with symbol m (and continuous frame multiplier if F, G are in addition continuous frames). Among continuous frame multipliers one can find time-frequency localization operators [19, 20] (also called short-time Fourier transform multipliers) and Calderón–Toeplitz operators [36, 37]. Recently, a notion of frame (and related topics such as bases, Bessel maps, Riesz bases, and Riesz-Fischer maps) in space of distributions is appeared in [46, 48], involving a rigged Hilbert space, or Gel’fand triplet, i.e. a triple D[t] ⊂ H ⊂ D × [t × ], where D[t] is a dense subspace of H endowed with a locally convex topology t, stronger than the one induced by the Hilbert norm and D × [t × ] is the conjugate dual of D[t] with the strong dual topology t × . If D[t] is reflexive, then the inclusions are dense and continuous. Analogous concepts in rigged Hilbert spaces and for the discrete case have been considered also in [14]. The aim of this paper is then to give a correspondent notion of multipliers of distribution maps. A preliminary, but very confined, study about distribution multipliers actually was given in [46].

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In contrast with a large part of the current literature, we will not pay attention to bounded multipliers only. To give an example of the importance of unbounded multipliers, we mention [4, 5] where they were used as tools to define non-selfadjoint Hamiltonians. Sufficient conditions for operators to be written as multipliers with a fixed sequence (or map) have been given in [12, 13, 25]. The paper is organized as follows. We start by recalling some preliminaries in Sect. 2. Then, in Sect. 3, we give the definitions of distribution multipliers. They can actually be formulated in two different ways, i.e. as operators from D to D × or as operators on H . Questions about density of domain and closedness of unbounded multipliers are discussed in Sect. 4, while Riesz distribution multipliers are studied and some results about symbolic calculus are obtained in Sect. 5.

2 Preliminary Definitions and Facts Throughout the paper H indicates a Hilbert space with inner product ·|· and norm

· . We denote by D(T ) and R(T ) the domain and the range of an operator T : D(T ) ⊂ H → H . If T is densely defined, then we write T ∗ for its adjoint. A sequence φ = {φn }n∈N ⊂ H is called total if f |φn = 0 for every n ∈ N implies that f = 0. In particular, discrete frames are total sequences. A Riesz basis φ is a total sequence satisfying for some A, B > 0

A

 n∈N

42 4 4 4  4 4 |cn | ≤ 4 cn φn 4 ≤ B |cn |2 , 4 4 2

n∈N

∀{cn } ∈ 2 (N),

n∈N

where 2 (N) is the usual space of square integrable complex sequences. Let D[t] be a dense subspace of H endowed with a locally convex topology t, stronger than the topology induced by the Hilbert norm. The vector space of all continuous conjugate linear functionals on D[t] (the conjugate dual of D[t]) is denoted by D × [t × ], and is endowed with the strong dual topology t × , defined by the seminorms qM (F ) = sup |F |g |, g∈M

F ∈ D ×,

where M is a bounded subsets of D[t]. With a well-known identification procedure (see [32]), H is considered as subspace of D × [t × ]. The triplet D[t] ⊂ H ⊂ D × [t × ] is called rigged Hilbert space or Gel’fand triplet [26, 27]. If D[t] is reflexive H is continuously and densely embedded in D × [t × ]. Denoting by #→ the continuous and dense embedding, the triple is also denoted by

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D[t] #→ H #→ D × [t × ]. In this way, the sesquilinear form B(·, ·) which puts D and D × in duality extends the inner product of H , i.e. B(ξ, η) = ξ |η , for every ξ, η ∈ D: we adopt the symbol ·|· for both of them. Let us denote by L (D, D × ) the vector space of all continuous linear maps from D[t] into D × [t × ] [2]. If D[t] is reflexive, it is possible to introduce an involution X → X† in L (D, D × ) by the identity: X† η|ξ = Xξ |η ,

∀ξ, η ∈ D.

Hence, in this case, L (D, D × ) is a † -invariant vector space. If D[t] is a smooth space (e.g. Fréchet and reflexive), then L (D, D × ) is a quasi *-algebra over L † (D) (Definition 2.1.9 of [2]). We also denote by L (D) the algebra of all continuous linear operators Y : D[t] → D[t] and by L (D × ) the algebra of all continuous linear operators Z : D × [t × ] → D × [t × ]. If D[t] is reflexive, for every Y ∈ L (D) there exists a unique operator Y × ∈ L (D × ), the adjoint of Y , such that F |Y g = Y × F |g ,

∀F ∈ D × , g ∈ D.

In similar way an operator Z ∈ L (D × ) has an adjoint Z × ∈ L (D) such that (Z × )× = Z. We denote by L † (D) the algebra of all closable operators A in H such that D(A) = D, D(A∗ ) ⊇ D, and A, A∗ leave D invariant. With the involution A → A∗ D = A† , L † (D) is a *-algebra. In this paper (X, μ) denotes a measure space with a σ -finite positive measure μ. We recall that a measurable set A ⊆ X is called an atom if μ(A) > 0 and for every B ⊆ A we have either μ(B) = 0 or μ(B) = μ(A). A measure space (X, μ) is called atomic if there exists a partition {An }n∈N of X consisting of atoms and sets of measure zero. We write L1 (X, μ), L2 (X, μ), and L∞ (X, μ) for the usual spaces of (classes of) measurable functions. Moreover, m ∞ denotes the essential supremum of m ∈ L∞ (X, μ). For simplicity, we write L1 (R), L2 (R), and L∞ (R) when we assume the Lebesgue measure. The Fourier transform of f ∈ L1 (R) is defined as f (γ ) = R f (x)e−2π iγ x dx and it extends to a unitary operator of L2 (R) in a standard way. In this paper we consider weakly measurable maps: given a measure space (X, μ) with μ a σ -finite positive measure, ω : x ∈ X → ωx ∈ D × is a weakly measurable map if, for every f ∈ D, the complex valued function x → f |ωx is μ-measurable. If not otherwise specified, throughout the paper we will work with a fixed rigged Hilbert space D[t] ⊂ H ⊂ D × [t × ] with D[t] reflexive and a measure space (X, μ) as described before. We start by recalling simple definitions about weakly measurable maps. Since the form which puts D and D × in conjugate duality is an extension of the inner product of H , we write f |ωx for ωx |f , f ∈ D. Definition 1 ([46, Definition 2.2]) Let ω : x ∈ X → ωx ∈ D × be a weakly measurable map, then:

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107

1. ω is total if f ∈ D and f |ωx = 0 μ-a.e. x ∈ X implies f = 0; 2. ω  is μ-independent if the unique μ-measurable function ξ : X → C such that: X ξ(x)g|ωx dμ = 0, for every g ∈ D, is ξ(x) = 0 μ-a.e. Definition 2 ([46, Definition 3.2]) A weakly measurable  map ω is a Bessel distribution map (briefly: Bessel map) if for every f ∈ D, X |f |ωx |2 dμ < ∞. It is convenient to consider D[t] as a Fréchet space because of the following proposition. Proposition 1 ([46, Proposition 3.1]) Let D[t] be a Fréchet space and ω : x ∈ X → ωx ∈ D × a weakly measurable map. The following statements are equivalent. 1. ω is a Bessel map. 2. There exists a continuous seminorm p on D[t] such that  1/2 2 |f |ωx | dμ ≤ p(f ), ∀f ∈ D. X

3. For every bounded subset M of D there exists CM > 0 such that ! ! ! ! ξ(x)ωx |f dμ! ≤ CM ξ 2 , ∀ξ ∈ L2 (X, μ). sup ! f ∈M

X

As a consequence of the previous proposition, we have [46]: • the conjugate linear functional on D:  Λξω := ξ(x)ωx dμ X

is defined in weak sense, and is continuous, i.e. Λω ∈ D × [t × ]; ξ • the synthesis operator Tω : L2 (X, μ) → D × [t × ] defined by Tω : ξ → Λω is continuous; • the analysis operator Tω× : D[t] → L2 (X, μ) defined by (Tω× f )(x) = f |ωx is continuous; • the frame operator Sω : D[t] → D × [t × ], Sω := Tω Tω× is continuous, i.e. Sω ∈ L (D, D × ). ξ

We will often work with a special class of Bessel maps which is defined as follows. Definition 3 ([46, Definition 3.2]) A Bessel distribution map ω is called bounded Bessel map if there exists B > 0 such that  |ωx |f |2 dμ ≤ B f 2 , X

∀f ∈ D.

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If ω is a bounded Bessel map, with a limit procedure, we have [46]: • Λω is bounded in D( · ), then it has a bounded extension Λ˜ ω to H √ ; • the synthesis operator Tω has range in H , it is bounded and Tω ≤ B; • the (Hilbert adjoint) operator Tω∗ : H → L2 (X, μ) extends Tω× and Tω∗ f = η, where η is the limit in L2 (X, μ) of the functions Tω× fn : x → fn |ωx , where {fn } is a sequence in D converging to f (we will also denote the function Tω∗ f by x → f |ωˇ x for f ∈ H , i.e. we consider ωˇ x as an “extension” of the linear functional ωx ); • the operator

Sω = Tω Tω∗ is bounded and it is an extension of Sω . ξ

ξ

Now it is time to recall a notion of frames in the distribution context (Definition 3.6 of [46]). Definition 4 ([46, Definition 3.6]) Let D[t] ⊂ H ⊂ D × [t × ] be a rigged Hilbert space, with D[t] a reflexive space and ω a weakly measurable map. We say that ω is a distribution frame if there exist A, B > 0 (called frame bounds) such that  A f 2 ≤

|f |ωx |2 dμ ≤ B f 2 ,

∀f ∈ D.

X

Moreover, we say that • ω is a tight distribution frame if we can choose A = B as frame bounds of ω; • ω is a Parseval distribution frame if A = B = 1 are frame bounds of ω. If ω is a distribution frame, then the frame operator Sˆω satisfies the inequalities A f ≤ Sˆω f ≤ B f ,

∀f ∈ H .

Since Sˆω is symmetric, this implies that Sˆω has a bounded inverse Sˆω−1 everywhere defined in H . A Parseval distribution frame satisfies Definition 4 with Sω = ID , the identity operator of D, and

Sω = IH , the identity operator of H . Example 1 ([46, Example 3.18]) Let us consider the rigged Hilbert space S (R) ⊂ L2 (R) ⊂ S × (R), where S (R) is the Schwartz space of rapidly decreasing C ∞ -functions on R and the conjugate dual S × (R) is the space of tempered distributions. Let δ be the weakly measurable map δ : x ∈ R → δx ∈ S × (R), where δx stands for the δ distribution centered at x. As known, δx acts in the following way δx |φ = φ(x), for every φ ∈ S (R). Then one trivially has 

 |φ|δx | dx = 2

R

R

|φ(x)|2 dx = φ 2 ,

∀φ ∈ S (R),

where φ is the norm in L2 (R); hence, δ is a Parseval frame.

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We note that this example is based on the measure space (X, μ) = (R, λ), where λ is the Lebesgue measure on R, and of course it is not a continuous frame. On the contrary, the definition of distribution frame reduces to that of discrete frame when (X, μ) is the set N with the counting measure γ . Indeed we have the following. Proposition 2 Let ω : n ∈ N → ωn ∈ D × be a bounded Bessel distribution (resp., distribution frame) on D[t] ⊂ H ⊂ D × [t × ]. Then {ωn }n∈N ⊂ H and {ωn }n∈N is a Bessel sequence (resp., frame) of H . Proof Assume that {ωn }n∈N is a bounded Bessel distribution map and fix m ∈ N. The linear functional f → f |ωm for f ∈ D is bounded with respect to the norm of H , because  |f |ωn |2 ≤ B f 2 , ∀f ∈ D. (2) |f |ωm |2 ≤ n∈N

This means that ωm ∈ H . A standard argument ([16, Lemma 5.1.9]) shows that (2) extends for each f ∈ H , i.e. {ωn }n∈N is a Bessel sequence. If {ωn }n∈N is a distribution frame, then the conclusion follows in a similar way.   Definition 5 Let ω, θ be distribution frames. We say that θ is a dual frame of ω if  f |θx ωx |g dμ,

f |g =

∀f, g ∈ D.

X

To formulate the following result, we recall that there exists a unique operator Rω ∈ L (D) such that Sω Rω f = f for every f ∈ D [46, Lemma 3.8]. Proposition 3 ([46, Proposition 3.10]) Let ω be a distribution frame with frame bounds A and B. Then the map θ : X → D × defined by θx := Rω× ωx for x ∈ X is a distribution frame with bounds B −1 and A−1 and it is a dual frame of ω. The map θ in Proposition 3 is called the canonical dual frame of ω. Definition 6 ([46, Definition 2.3]) Let D[t] be a locally convex space, D × its conjugate dual and ω : x ∈ X → ωx ∈ D × a weakly measurable map. Then ω is a distribution basis for D if, for every f ∈ D, there exists a unique μ-measurable function ξf such that:  ξf (x)ωx |g dμ,

f |g =

∀f, g ∈ D

X

and, for every x ∈ X, the linear functional f ∈ D → ξf (x) ∈ C is continuous in D[t].

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Given a distribution basis ω we can simply write in weak sense  f =

ξf (x)ωx dμ,

∀f ∈ D.

X

Moreover, ω is μ-independent. Since f ∈ D → ξf (x) continuously, there exists a unique weakly μ-measurable map θ : X → D × such that: ξf (x) = f |θx for every f ∈ D. We call θ dual map of ω. If θ is μ-independent, then it is a distribution basis too. The next two notions are the counterparts of Riesz and orthonormal bases of the discrete context, which are particular cases of Definition 6. Definition 7 ([46]) A weakly measurable map ω : X → D × is a Riesz distribution basis if it is a μ-independent distribution frame. A weakly measurable map ζ : X → D × is Gel’fand distribution basis if it is a μ-independent Parseval distribution frame. In a way similar to the discrete case, we can give some equivalent conditions for a Bessel distribution map to be a Riesz distribution basis. Proposition 4 ([46, Proposition 3.19]) Let D ⊂ H ⊂ D × be a rigged Hilbert space and let ω : x ∈ X → ωx ∈ D × be a Bessel distribution map. Then the following statements are equivalent. 1. ω is a Riesz distribution basis; 2. if ζ is a Gel’fand distribution basis, then the operator W defined, for f ∈ H , by 



f =

ξf (x)ζx dμ → Wf = X

ξf (x)ωx dμ X

is continuous and has bounded inverse; 3. the synthesis operator Tω is a topological isomorphism of L2 (X, μ) onto H ; 4. ω is total and there exist A, B > 0 such that 4 42 4 4 4 ≤ B ξ 2 , A ξ 22 ≤ 4 ξ(x)ω dμ x 2 4 4

∀ξ ∈ L2 (X, μ).

X

If ω is a Riesz distribution basis with frame bounds A, B, then ω possesses a unique dual frame θ (so the canonical dual frame) which is also a Riesz distribution basis with frame bounds B −1 and A−1 (see [46, Proposition 3.20]). In particular, a Gel’fand distribution basis ζ coincides with its dual basis. Example 2 ([46, Example 3.18]) Let us come back to Example 1. The distribution frame δ : x ∈ R → δx ∈ S × (R) is clearly λ-independent, then δ is a Gel’fand distribution frame. In Proposition 2 we made a consideration about the case (X, μ) = (N, γ ) and discrete frames. Now we compare distribution frames with continuous frames.

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There is indeed a remarkable difference about the possibility to define “Riesz maps”. To explain the difference in details, we recall that a family {Fx }x∈X ⊂ H is a Riesz continuous map if one of the following statements holds (see [3, 24]) 1. {Fx }x∈X is a continuous frame and the operator CF : H → L2 (X, μ) defined by (CF f )(x) = f |Fx is surjective; 2. {F  x }x∈X is a continuous frame and μ-linearly independent, i.e. if c : X → C and X c(x)f |Fx dμ(x) = 0 for all f ∈ H , then c(x) = 0 μ-almost everywhere. It was proven in [35, Corollary 4.3] and in [39, Theorem 9] that Riesz continuous maps can be defined only if the space (X, μ) is atomic. In contrast, Riesz distribution maps can be defined with non-atomic measure spaces (X, μ) (as we have seen in Example 1). Note that the formulations of Riesz continuous map and Riesz distribution map are totally analogue.

3 Distribution Multipliers Let D[t] ⊂ H ⊂ D × [t × ] be a rigged Hilbert space. In the distribution context, there is more than one way to define multipliers. The first way we describe consists of operators acting on D[t] with values in D × [t × ]. We suppose that D[t] is a reflexive Fréchet space and (X, μ) is a measure space with μ a σ -finite positive measure. Let ω, θ : X → D × be two weakly measurable Bessel maps and m ∈ L∞ (X, μ). Then the sesquilinear form  Ωm,ω,θ (f, g) :=

m(x)f |ωx θx |g dμ X

is defined for all f, g ∈ D. By Proposition 1(ii) we have |Ωm,ω,θ (f, g)| ≤ m ∞ f |ωx 2 θx |g 2 ≤ m ∞ p(f )p(g) for all f, g ∈ D. This means that Ωm,ω,θ is jointly continuous on D[t] and then there exists an operator Mm,ω,θ ∈ L (D, D × ), such that Mm,ω,θ f |g = Ωm,ω,θ (f, g),

∀f, g ∈ D.

For brevity we write  Mm,ω,θ f =

m(x)f |ωx θx dμ,

∀f ∈ D

X

and we call Mm,ω,θ the outer distribution multiplier of ω and θ with symbol m.

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As in [8], we have, if Tω× and Tθ are the analysis and synthesis operators of ω and θ , respectively, Dm : L2 (X, μ) → L2 (X, μ) is the multiplication by m defined by † = Mm,θ,ω . Dm f (x) := m(x)f (x), then Mm,ω,θ = Tθ Dm Tω× . Moreover, Mm,ω,θ Now we move to the second way to define multipliers, namely operators acting on H . Again we consider a rigged Hilbert space D[t] ⊂ H ⊂ D × [t × ], a measure space (X, μ) with μ a σ -finite positive measure, but the choice of ω, θ and m is more general. Indeed, let ω, θ be two weakly measurable maps and m : X → C a μ-measurable function. Let us define the subspace D(Mm,ω,θ ) of f ∈ D such that the integral  m(x)f |ωx θx |g dμ X

is convergent for all g ∈ D and the linear functional  g →

m(x)f |ωx θx |g dμ

(3)

X

is bounded with respect to the norm of H . An operator Mm,ω,θ : D(Mm,ω,θ ) → H can be defined as follows: for every f ∈ D(Mm,ω,θ ), Mm,ω,θ f is the unique element in H associated with the functional (3) by the Riesz lemma, i.e.  Mm,ω,θ f |g =

m(x)f |ωx θx |g dμ,

∀f ∈ D(Mm,ω,θ ), g ∈ D.

X

 For shortness, we write Mm,ω,θ f = X m(x)f |ωx θx and call Mm,ω,θ the distribution multiplier of ω and θ with symbol m. If ω and θ are Bessel distribution maps (resp. Gel’fand bases, Riesz distribution bases, distribution frames), then Mm,ω,θ is called a Bessel distribution (resp. Gel’fand distribution, Riesz distribution, distribution frame) multiplier. In the language of representation of sesquilinear forms [34, 38], we can say that Mm,ω,θ is the operator associated with Ωm,ω,θ . In the discrete setting, operators associated with sesquilinear forms induced by sequences have been studied in [5, 17, 18]. We first pay attention to distribution multipliers defined (resp. bounded) on D. Proposition 5 Let m ∈ L∞ (X, μ). 1. If ω is a Bessel distribution map and θ is a bounded Bessel distribution map, then Mm,ω,θ is a well-defined operator Mm,ω,θ : D → H . 2. If ω and θ are bounded Bessel distribution maps with bound Bω , Bθ , respectively,

then Mm,ω,θ is bounded √ and it extends to a bounded operator Mm,ω,θ on H with

m,ω,θ ≤ Bω Bθ m ∞ . norm M

m,ω,θ = Tθ Dm Tω∗ and Mm,ω,θ ∗ = 3. If ω, θ are bounded Bessel maps, then M

m,ω,θ . M

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The proof is an adaptation of [10, Lemma 3.3] and it is omitted. However, we want to make the following remark about Proposition 5. Remark 1 Let m ∈ L∞ (X, μ) and ω, θ Bessel distribution maps, then Mm,ω,θ is not necessarily bounded in the norm of H . Indeed let us consider S (R) ⊂ L2 (R) ⊂ f |xδx = S × (R) and ω : R → S × (R) defined by ωx = xδx , i.e. the distributions  xf (x) for f ∈ S (R). Then ω is a Bessel distribution map since R |f |xδx |2 dx =  2 R |xf (x)| dx is finite for all f ∈ S (R). Let θ be the distribution frame given by θx = δx and m(x) = 1 for x ∈ R, then Mm,ω,θ is defined on f ∈ S (R) and (Mm,ω,θ f )(x) = xf (x) for x ∈ R. Clearly, Mm,ω,θ is not bounded. Multipliers with a bounded inverse defined on the whole space have a special interest, since they lead to reconstruction formulas, as shown in the discrete case in [9, 43, 44]. In the following result we show how reconstruction formulas can be found by a distribution multiplier having a right or left inverse. Theorem 1 Let ω, θ : X → D × be weakly measurable maps and m : X → C such that the distribution multiplier Mm,ω,θ is defined on D. 1. If there exists J ∈ L (D) such that Mm,ω,θ Jf = f for every f ∈ D, then the weakly measurable map ρ : X → H defined by ρx = J † (m(x)ωx ) satisfies  ∀f, g ∈ D. f |g = f |ρx θx |g dμ, X

L (D × )

2. If there exists K ∈ such that KMm,ω,θ f = f for every f ∈ D the weakly measurable map τ : X → H defined by τx = K(m(x)θx ) satisfies  ∀f, g ∈ D. f |g = f |ωx τx |g dμ, X

Proof 1. Let J ∈ L (D) as in the statement. Then



f |g = Mm,ω,θ Jf |g =

m(x)Jf |ωx θx |g dμ X



f |J † (m(x)ωx ) θx |g dμ,

= X

for all f, g ∈ D. 2. Let K ∈ L (D × ) as in the statement. Then



f |g = KMm,ω,θ f |g = Mm,ω,θ f |K g =

m(x)f |ωx θx |K † g dμ

†

X

 f |ωx K(m(x)θx )|g dμ,

= X

for all f, g ∈ D.

 

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Example 3 Let us consider again the rigged Hilbert space S (R) ⊂ L2 (R) ⊂ S × (R) and ω = θ the distribution frames defined by ωx = θx = δx for every x ∈ R and m ∈ C ∞ (R) a function such that 0 < infx∈R |m(x)| ≤ supx∈R |m(x)| < ∞. The multiplier Mm,ω,θ is of course defined on S (R) and Mm,ω,θ f = mf . Clearly, the operator J : S (R) → S (R) defined by Jf = m−1 f belongs to L (S (R)) and the operator K : S × (R) → S × (R) defined by KF = m−1 F belongs to L (S × (R)). Moreover, KMm,ω,θ f = Mm,ω,θ Jf = f for every f ∈ S (R). The reconstruction formulas in Theorem 1 hold in particular with ρx = τx = δx . In Sect. 4 we will give conditions for a Riesz multiplier to be invertible with bounded inverse.

4 Unbounded Distribution Multipliers In Proposition 5 we gave a condition for a distribution multiplier to be bounded. Not only bounded multipliers are interesting, of course; we refer to [4, 5] where unbounded discrete multipliers have been studied in the context of non-self-adjoint Hamiltonians. So in this section we want to analyze some aspects of unbounded distribution multipliers. Assuming that D[t] is a reflexive Fréchet space, ω, θ are Bessel distribution maps and m ∈ L∞ (X, μ), we easily see that Mm,ω,θ is actually a restriction of Mm,ω,θ . Indeed, D(Mm,ω,θ ) = {f ∈ D : Mm,ω,θ f ∈ H } and Mm,ω,θ = Mm,ω,θ |D(Mm,ω,θ ) . This fact leads, for instance, to the following conclusions. Proposition 6 Let D[t] be a reflexive Fréchet space. Let ω, θ be Bessel distribution maps and m ∈ L∞ (X, μ). If Mm,ω,θ : D → D × is bijective with a bounded inverse, then 1. Mm,ω,θ : D(Mm,ω,θ ) → H is bijective, densely defined and has a bounded inverse (consequently Mm,ω,θ is closed); 2. Mm,ω,θ ∗ = Mm,θ,ω . Proof 1. The fact that Mm,ω,θ : D(Mm,ω,θ ) → H is bijective follows easily since Mm,ω,θ is a restriction of Mm,ω,θ . Since the inclusions D[t] ⊂ H ⊂ D × [t × ] are continuous • there exists a continuous seminorm p on D[t] and f ≤ p(f ) for all f ∈ D; • for all continuous seminorms q on D × [t × ] there exists αq > 0 such that q(f ) ≤ αq f for all f ∈ D. By hypothesis Mm,ω,θ −1 : D × → D is bounded, so there exists a continuous seminorm q on D × [t × ] such that for all continuous seminorms p on D[t] we have p(Mm,ω,θ −1 F ) ≤ q(F ) for all F ∈ D × [t × ]. Hence, for all h ∈ H

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Mm,ω,θ −1 h ≤ p(Mm,ω,θ −1 h) = p(Mm,ω,θ −1 h) ≤ q(h) ≤ αq h . Thus Mm,ω,θ has a bounded inverse. The continuity of Mm,ω,θ −1 implies also that D(Mm,ω,θ ) is dense, because D(Mm,ω,θ ) is the inverse image of H which is dense in D × [t × ]. 2. Clearly, Mm,ω,θ ∗ is bijective and Mm,θ,ω ⊆ Mm,ω,θ ∗ . The conclusion of point † (1) holds also for Mm,θ,ω since Mm,θ,ω coincides with a restriction of Mm,ω,θ = ∗ Mm,θ,ω . Thus we can conclude that Mm,θ,ω = Mm,ω,θ .   Even though Mm,ω,θ is not necessarily bounded, Proposition 6 makes use of boundedness of m. This may be a strong hypothesis, thus we now look for less restrictive assumptions to ensure that Mm,ω,θ is densely defined. In the discrete context it is very easy to prove that a multiplier Mm,φ,ψ of a Hilbert space H , where φ = {φn }n∈N is a Riesz basis and ψ = {ψn }n∈N a sequence of H , is densely defined whatever the symbol m = {mn }n∈N is. Indeed, there exists  = {φ n } biorthogonal to φ, a unique total sequence (in particular a Riesz basis) φ m |φn = δm,n (the Kronecker symbol) for all m, n ∈ N. Thus D(Mm,φ,ψ ) is i.e. φ . dense, because it contains φ On the contrary, when D ⊂ H ⊂ D × is a rigged Hilbert space and ω : X → D × is a Riesz distribution basis, then we may not find a function ρ : X → D which is biorthogonal to ω in the sense that  ρy |ωx =

1 if x = y, 0 if x = y.

Indeed, let us consider the Riesz distribution basis given by the Dirac deltas ωx = δx , x ∈ R, on the rigged Hilbert space S (R) ⊂ L2 (R) ⊂ S (R)× (Example 1). Then there is no x ∈ R and f ∈ S (R) such that f (x) := f |δx = 1 and f (z) := f |δz = 0 for all z = x. Thus we have to manage a new problem in order to study densely defined distribution multipliers. Taking again the example of ωx = δx , x ∈ R, on S (R) ⊂ L2 (R) ⊂ S (R)× , we note that for any symbol m : R → C and for θ = ω the multiplier Mm,ω,θ is densely defined. We will give the proof in Theorem 2 in a more general context. Here we confine ourselves to give the following remark. Note that we say that a subset V of a Hilbert space H is total if f |h = 0 for all f ∈ V implies h = 0 (then the linear span of V is dense in H ). Remark 2 Let λ be the Lebesgue measure on R. Let α : R → C be a positive λmeasurable function and define Vα := {f ∈ C0∞ (R) : |f (x)| ≤ α(x), x ∈ R}. We prove that the subset Vα is total in L2 (R) dividing the proof into three steps. First of all, α is locally bounded away from zero in a.e. x ∈ R, i.e. B c := R\B is measurable with measure zero, where B = {x ∈ R : there exists an interval Ux ⊂ R of x such that essinfy∈Ux α(y) > 0}.

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Indeed, fix n ∈ N. For every x ∈ B c , there exists a measurable set Un,x containing x such that essinfy∈Un,x α(y) ≤ n1 . Then B c ⊆ ∪x∈B c Un,x ⊆ {y ∈ R : α(y) ≤ n1 }, thus the outer measure λo (B c ) ≤ λ({y ∈ R : α(y) ≤ n1 }) → 0 for n → ∞. Hence B c is measurable with measure zero. Now, if x ∈ B and mx = essinfy∈Ux α(y) > 0, then mx χI ∈ Vα where χI is the characteristic function of any interval I ⊂ Ux . Taking into account that B c has measure zero, we conclude that there is a subset V total in L2 (R) such that every f ∈ V satisfies f (x) = mχI (for some m > 0 and an interval I ) and 0 ≤ f (x) ≤ α(x) for every x ∈ R. Finally, every function f ∈ V can be approximated with C0∞ functions {fk } with 0 ≤ fk ≤ f . Hence, Vα is total. The reason of talking about the example ωx = δx is that it suggests to consider variations of the condition of biorthogonality as stated in the next definitions. For a better comparison, we rewrite a property of Vα (with α : R → C a positive measurable function) in the following way: for every f ∈ Vα there exists a bounded subset Xf ⊂ R and |f |δx | ≤ α(x) for x ∈ Xf and f |δx = 0 for x ∈ / Xf . Definition 8 Let D ⊂ H ⊂ D × be a rigged Hilbert space and ω : X → D × a weakly measurable function. We say that 1. ω is pseudo-orthogonal if there exists a subset V ⊂ D total in H such that for every f ∈ V there exists a measurable subset Xf ⊂ X with μ(Xf ) < ∞, supx∈Xf |f |ωx | < ∞ for x ∈ Xf and f |ωx = 0 for x ∈ / Xf ; 2. ω is hyper-orthogonal if for every positive measurable function α : X → C there exists a subset Vα ⊂ D total in H such that for every f ∈ Vα there exists a measurable subset Xf ⊂ X with μ(Xf ) < ∞, |f |ωx | ≤ α(x) for x ∈ Xf and f |ωx = 0 for x ∈ / Xf . Note that these definitions are covered by a Riesz (discrete) basis {φn }n∈N (more generally by a sequence {φn }n∈N having a total biorthogonal sequence {ψn }n∈N , i.e. φn |ψm = δn,m ). Furthermore, if ω is hyper-orthogonal, then it is also pseudoorthogonal. We are now able to formulate results about the density of domains of distribution multipliers. We denote by L2loc (X, μ) the space of measurable functions f on X such that f ∈ L2 (U ) for every bounded measurable subset U ⊆ X. Theorem 2 Let D[t] ⊂ H ⊂ D × [t × ] be a rigged Hilbert space, ω : X → D × a weakly measurable function, θ : X → D × a bounded Bessel distribution map with Bessel bound Bθ , and m : X → C a μ-measurable function. 1. If ω is pseudo-orthogonal and m ∈ L2loc (X, μ), then the distribution multiplier Mm,ω,θ is densely defined. 2. If ω is a bounded Bessel distribution map and hyper-orthogonal, then the distribution multiplier Mm,ω,θ is densely defined.

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Proof 1. Let V and Xf be as in Definition 8(1). For f ∈ V , g ∈ D we have by Cauchy– Schwarz inequality that ! !  ! ! ! m(x)f |ωx θx |g dμ! ≤ C ! ! X

Xf

1

|m(x)||θx |g |dμ ≤ CBθ2 m L2 (Xf ) g ,

where C = supx∈Xf |f |ωx |. Thus f ∈ D(Mm,ω,θ ), and consequently D(Mm,ω,θ ) is dense because V is total. 2. The proof is divided into three parts. If m ∈ L∞ (X, μ), then the conclusion follows by Proposition 5 since D(Mm,ω,θ ) = D. If |m(x)| ≥ 1 a.e., then we take α(x) = |m(x)|−1 a.e. in Definition 8 (item 2). Thus there exists Vα ⊂ D total in H such that for every f ∈ Vα there exists a compact subset Xf ⊂ X and ! !  ! ! ! m(x)f |ωx θx |g dμ! ≤ ! ! X

1

Xf

1

|m(x)||m(x)|−1 |θx |g |dμ ≤ μ(Xf ) 2 Bθ2 g ,

for every g ∈ D (the last inequality is due to Cauchy–Schwarz inequality). This means that D(Mm,ω,θ ) is dense. Finally, let m be a generic measurable function. Then it is possible to write m as sum of two measurable functions m = m1 + m2 such that m1 ∈ L∞ (X) and |m2 | ≥ 1. Then Mm1 ,ω,θ is well-defined on D and Mm2 ,ω,θ is defined on a subspace of D dense in H . As consequence, also Mm,ω,θ = Mm1 ,ω,θ + Mm2 ,ω,θ is densely defined.   We show other examples of weakly measurable maps satisfying Definition 8. Example 4 For x ∈ R consider the function ωx defined by ωx (y) := e−2π ixy for y ∈ R. Then ωx is a distribution on L1 (R) ∩ L2 (R) and in Example 3.17 of [46] it was proved that ω is a distribution frame. Choosing V = {f ∈ L2 (R) : f ∈ C0∞ (R)}, where f denotes the Fourier transform of f , in Definition 8, we conclude that ω is pseudo-orthogonal. Now let α : R → C be a positive measurable function and define Vα = {f ∈ L2 (R) : f ∈ C0∞ (R)and|f (x)| ≤ α(x), x ∈ R}. By the considerations in Remark 2 and since the Fourier transform is unitary in L2 (R), the set Vα is total, i.e. ω is also hyper-orthogonal. Example 5 Let D[t] be a dense subspace of L2 (R) endowed with a locally convex topology t, stronger than the topology of L2 (R), and such that C0∞ (R) ⊂ D. Let g ∈ L2 (R) have support in a bounded interval I . Define ωx (t) = g(t − x) for t ∈ R. Then the weakly measurable map ω : R → L2 (R) is pseudo-orthogonal (again one can take V = C0∞ (R)). We conclude this section by turning the attention to a sufficient condition for a distribution multiplier to be closable.

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Proposition 7 Let ω, θ : X → D × be weakly measurable functions and m : X → C be a μ-measurable function. If Mm,θ,ω is densely defined (in particular if ω, θ are bounded Bessel distribution maps and θ is hyper-orthogonal), then Mm,ω,θ is closable. Proof For f ∈ D(Mm,ω,θ ) and g ∈ D(Mm,θ,ω ) we have Mm,ω,θ f |g =   f |Mm,θ,ω g . This means that Mm,ω,θ ⊆ (Mm,θ,ω )∗ , i.e. Mm,ω,θ is closable.

5 Riesz Distribution Multipliers In this section, we examine the case where ω and θ are Riesz distribution bases, reconsidering the Examples 4.1 and 4.2 of [46]. Let ω and θ be distribution Riesz bases, ωˇ and θˇ their extensions to H with the limit procedure described just after Definition 3, and a μ-measurable function m : X → C such that the integral: 

m(x)f |ωˇ x θˇx |g dμ

X

is convergent for all f, g ∈ H . Since θ is a Riesz basis, the analysis operator Tθ∗ is a topological isomorphism of H onto L2 (X, μ), then Tθ∗ (H ) = L2 (X, μ). It follows that m(x)f |ωˇ x ∈ L2 (X, μ) for all f ∈ H . Furthermore, the operator Mm,ω,θ : D → H  Mm,ω,θ, f =

m(x)f |ωx θx dμ,

∀f ∈ D,

X

is well-defined. Analogously, for all g ∈ H one has m(x)g|θˇx ∈ L2 (X, μ) and † the operator Mm,ω,θ :D →H  † Mm,ω,θ g := Mm,θ,ω g =

m(x)g|θx ωx dμ

∀g ∈ D

X

is well-defined. One has † Mm,ω,θ f |g = f |Mm,ω,θ g ,

∀f, g ∈ D.

Then Mm,ω,θ is a closable operator in H . It is not difficult to show that the domain of the closure D(M m,ω,θ ) is {f ∈ H : X |m(x)f |ωˇ x |2 dμ < ∞}. In general, the operators Mm,ω,θ are unbounded, so their product is not always defined. However, if they belong to the space L † (D), they can be multiplied. In the following example, some cases of unbounded multipliers in L † (D) are considered.

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Example 6 Let us consider the rigged Hilbert space S (R) ⊂ L2 (R) ⊂ S (R)× . We write f and fˇ for the Fourier transform and inverse Fourier transform of f ∈ S (R), respectively. Define: ωx = δx and θx (y) := e−2π ixy for y ∈ R, thus δx , θx ∈ S (R)× . Let OM (R) be the space of C ∞ -functions which, together with their derivatives, are polynomially bounded (see [32]). If m ∈ OM (R), for f ∈ S (R) we have ˇ ∗ fˇ, Mm,θ,ω f = mf and Mm,θ,θ f = m ˇ ∗ f. Mm,ω,ω f = mf, Mm,ω,θ f = m The above considerations lead in particular to the case of a Riesz basis ω and its dual θ of Example 4.2 [46]. To simplify the notation, we denote the multiplier as Mm := Mm,θ,ω . In Example 4.2 [46], it is shown that m(x) is a generalized eigenvalue of Mm , i.e.: † × ) θx |g = m(x)θx |g , ∀g ∈ D, (Mm

μ-a.e.

x ∈ X,

μ-a.e.

x ∈ X.

† and m(x) is a generalized eigenvalue of Mm , i.e.:

(Mm )× ωx |g = m(x)ωx |g , ∀g ∈ D, A consequence is the following:

Proposition 8 Let Mm1 and Mm2 be multipliers of a Riesz basis ω and its dual θ , such that Mmi ∈ L † (D), i = 1, 2. Then Mm1 Mm2 = Mm1 m2 . Proof For all f, g ∈ D  † (Mm2 Mm1 )f |g = Mm1 f |Mm g = 2

X

† m1 (x)f |ωx θx |Mm g dμ = 2

 = X

m1 (x)m2 (x)f |ωx θx |g dμ = Mm1 m2 f |g  

and the proof is completed.

† † † † † Analogously, Mm 1 and Mm2 can be multiplied, and (Mm1 Mm2 ) = Mm2 Mm1 . This † shows that the set of multipliers in L (D) of dual Riesz bases is a †-subalgebra in −1 and M are defined in L † (D), by Proposition 8, L † (D). Furthermore, if both Mm 1 m

−1 = M . The considered case of multipliers in L † (D) allows one has that Mm 1 m

to handle easily the symbolic calculus, but operators in L † (D) are, in general, unbounded. What can be said about the case of bounded operators? Obviously, if m ∈ L∞ (X, μ) and ω and θ are Riesz bases, Mm is bounded. Viceversa, the following proposition holds: Proposition 9 Let ω be a distribution Riesz basis with dual θ . If the multiplier Mm,ω,θ is bounded, then m ∈ L∞ (X, μ).

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Proof The proposition is true for the diagonal operator Am,ζ , i.e. a multiplier of the Gel’fand basis: ω = θ = ζ (see Example 4.1 of [46]). The same holds for Mm,ω,θ : in fact Mm,ω,θ and Am,ζ are similar via W of b) of Proposition 4, i.e. M m,ω,θ = W Am,ζ W −1 (see Example 4.2 of [46]), where M m,ω,θ and Am,ζ are their closure.   For conditions of invertibility of a multiplier, we can state the following in a more general form. Proposition 10 Let ω, θ : X → D × be weakly measurable maps and m(x) = 0 μ-a.e. in X. Then 1. If ω is μ-independent and θ is total, then Mm,ω,θ is injective. 2. If ω is total and θ is μ-independent, then Mm,ω,θ has dense range in H .  Proof Assume that Mm,ω,θ f = X m(x)f |ωx θx dμ = 0. Since θ is μindependent, we have m(x)f |ωx = 0 a.e., that is, f |ωx = 0 a.e.; but ω is total, then f = 0. For the range,  let g ∈ D such that Mm,ω,θ f |g = 0 for all f ∈ D, that is, Mm,ω,θ f |g = X m(x)f |ωx θx |g dμ = 0. Since ω is μ-independent, we have m(x)θx |g = 0 a.e. and g = 0, because θ is total.   In particular, if ω, θ are Riesz bases and m(x) is nonzero a.e., then Mm,ω,θ is invertible with densely defined inverse. To have a bounded inverse we can make use of an additional assumption. Proposition 11 Let Mm,ω,θ be a Riesz distribution multiplier. If there exists C > 0 such that 0 < C ≤ |m(x)| for all x ∈ X, the inverse of Mm,ω,θ is bounded. Proof By Proposition 10, the inverse exists. Let Aω , Bω and Aθ , Bθ be the lower and upper bounds of ω and θ , respectively. The operator Mm,ω,θ has closure

m,ω,θ := Tθ Dm Tω∗ . By Proposition 4, the operators Tω∗ , Tω , Tθ , T ∗ are extension M θ bounded, invertible with bounded inverses, so we have Aθ Aω C f ≤ Aθ C Tω∗ f 2 ≤ Tθ Dm Tω∗ f , and the proof is completed.

∀f ∈ D(Mm,ω,θ ),  

6 Conclusions Some questions about symbolic calculus in a more general set-up (not only in the case of dual Riesz bases) are open. For instance, it is known that in L (D, D × ) a partial multiplication is defined (see [2, 47]), thus a symbolic calculus may be developed for multipliers in L (D, D × ). The idea behind Definition 8 is connected to localization frames which were introduced in [29] and further studied (sometimes with variations) in [6, 7, 11, 22, 23, 28]. More precisely, some results in Sect. 4 can be extended considering a certain decay of x → f |ωx instead of assuming that f |ωx is null outside a bounded set.

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Acknowledgments R.C. has been partially supported by the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA-INdAM).

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24. J-P. Gabardo, D. Han, Frames associated with measurable spaces, Adv. Comput. Math., 18, (2003), 127–147. 25. L. Gavruta, ˇ Frames and operators, Appl. Comp. Harmon. Anal., 32 (2012), 139–144. 26. I.M. Gel’fand, G.E. Shilov, E. Saletan, Generalized Functions, Vol.III, Academic Press, New York, (1967). 27. I. M. Gel’fand, N. Ya. Vilenkin, Generalized Functions, Vol.IV, Academic Press, New York, (1964). 28. K. Gröchenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math., 18, (2003), 149–157. 29. K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10, (2004), 105–132. 30. K. Gröchenig, Representation and approximation of pseudodifferential operators by sums of Gabor multipliers, Appl. Anal., 90(3–4), (2010), 385–401. 31. C. Heil A Basis Theory Primer, Expanded Edition, Birkhäuser/Springer, New York, (2011). 32. J. Horvath, Topological Vector Spaces and Distributions, Addison-Wesley, 1966. 33. G. Kaiser, A friendly guide to wavelets, Birkhäuser, Boston, (1994). 34. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, (1966). 35. F. Li, P. Li, A. Liu, Decomposition of analysis operators and frame ranges for continuous frames, Numerical Functional Analysis and Optimization, 37, 2, (2016), 238–252. 36. R. Rochberg, Toeplitz and Hankel operators, wavelets, NWO sequences, and almost diagonalization of operators, Operator Theory: Operator Algebras and Applications, Part 1, (Proc. Symp. Pure Mathematics vol 51), (Providence, RI: American Mathematical Society), (1990), 425–444. 37. R. Rochberg, A correspondence principle for Toeplitz and Calderón-Toeplitz operators, Israel Math. Conf. Proc., 5, (1992), 229–243. 38. K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, Dordrecht, (2012). 39. M. Speckbacher, P. Balasz, Frames, their relatives and reproducing kernel Hilbert spaces, arxiv:1704.02818, (2017). 40. D. T. Stoeva, P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal., 33(2), (2012), 292–299. 41. D. T. Stoeva, P. Balazs, Canonical forms of unconditionally convergent multipliers, J. Math. Anal. Appl., 399(1), (2013), 252–259. 42. D. T. Stoeva, P. Balazs, Riesz bases multipliers, In M. Cepedello Boiso, H. Hedenmalm, M. A. Kaashoek, A. Montes-Rodríguez, and S. Treil, editors, Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation, vol 236 of Operator Theory: Advances and Applications, 475–482, Birkhäuser, Springer Basel, (2014). 43. D. T. Stoeva, P. Balazs, The dual frame induced by an invertible frame multiplier, in Sampling Theory and Applications (SampTA), 2015 International Conference on, IEEE, (2015), 101– 104. 44. D. T. Stoeva, P. Balazs, On the dual frame induced by an invertible frame multiplier, Sampling Theory in Signal and Image Processing, 15, (2016), 119–130. 45. D. T. Stoeva, P. Balazs, A survey on the unconditional convergence and the invertibility of multipliers with implementation, in Sampling - Theory and Applications (A Centennial Celebration of Claude Shannon), S. D. Casey, K. Okoudjou, M. Robinson, B. Sadler (Ed.), Applied and Numerical Harmonic Analysis Series, Springer, (2020). 46. C. Trapani, S. Triolo, F. Tschinke, Distribution Frames and Bases, J. Fourier Anal. Appl., 25, (2019), 2109–2140. 47. C. Trapani, F. Tschinke, Partial Multiplication of Operators in Rigged Hilbert, Int. Equ. Operator Theory, 51(4), (2005), 583–600. 48. F. Tschinke, Riesz-Fischer maps, Semiframes and Frames in rigged Hilbert spaces, arXiv:1910.14447, (2019).

Generalized Anti-Wick Quantum States Maurice de Gosson

Abstract Density operators are positive semidefinite operators with trace one representing the mixed states of quantum mechanics. The purpose of this contribution is to define and study a subclass of density operators on L2 (Rn ), which we call Toeplitz density operators. They correspond to quantum states obtained from a fixed function (“window”) by position-momentum translations, and reduce in the simplest case to the anti-Wick operators considered long ago by Berezin and extensively studied by Cordero and others. The rigorous study of Toeplitz operators requires the use of classes of functional spaces defined by Feichtinger. Keywords Density operator · Anti-Wick · Toeplitz · Modulation spaces

1 Introduction 1.1 Motivations A quantum mixed state on Rn consists of a sequence of pairs {(ψj , λj )}, where the ψj ∈ L2 (Rn ) are called pure states and the λj are probabilities summing up to one. Given a mixed state {(ψj , λj )} one defines its density operator as being the compact operator 

j λj Π (1) ρ

= j

j is the orthogonal projection in L2 (Rn ) onto the ray Cψj = on L2 (Rn ), where Π {αψj : α ∈ C}. The operator ρ

is positive semidefinite and has trace Tr(

ρ ) = 1,

M. de Gosson () Faculty of Mathematics, NuHAG, University of Vienna, Wien, Austria Institute of Mathematics, University of Würzburg, Würzburg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_7

123

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and one verifies using the spectral theorem that every positive semidefinite operator on L2 (Rn ) with trace unity can conversely be written in the form (1), and thus represents a mixed state [21]. The Wigner distribution of ρ

is, by definition, the function ρ ∈ L2 (R2n ) defined by ρ(z) =



(2)

λj W ψj (z),

j

where W ψj is the usual Wigner transform of ψj (convergence in the L2 -norm). In this work we will deal with a particular class of density operators and their generalizations to a continuous setting. Consider the ground state φz0 (x) = (π h¯ )−n/4 eip0 (x−x0 ) e−|x−x0 |

2 /2h

¯

(3)

of a linear oscillator H = 12 (|p − p0 |2 + |x − x0 |2 ) whose center z0 = (x0 , p0 ) is not known with precision. Suppose we have some partial information telling us that z0 is located somewhere on a lattice Λ ⊂ R2n consisting of a discrete set of phase space points zλ = (xλ , pλ ) and that there is a probability μzλ that the system

zλ the orthogonal projection on Cφzλ the is precisely in the state φzλ . Denoting by Π corresponding density operator 

ρ

=

zλ μzλ Π

(4)

zλ ∈Λ

has Wigner distribution ρ(z) =



μzλ W φzλ (z) .

(5)

zλ ∈Λ

Observing that W φzλ (z) = W φ0 (z − zλ ) this reduces to the Gabor-type [13] expansion ρ(z) =



μzλ W φ0 (z − zλ ) .

(6)

zλ ∈Λ

Let us now depart from the discrete case, and consider the somewhat more realistic situation where the center of the linear oscillator can be any point z in phase space R2n ; we assume the latter comes equipped with a certain Borel probability density μ. Formula (6) suggests that we define in this case a generalized Wigner distribution by  ρ(z) =



μ(z )W φ0 (z − z )d 2n z = (μ ∗ W φ0 )(z) .

(7)

Generalized Anti-Wick Quantum States

125

3 Notice that if one chooses for μ the atomic measure zλ ∈Λ μzλ δzλ , then (7) reduces to the discrete sum (5). It turns out that the operator ρ

obtained from ρ using the Weyl correspondence is a well-known mathematical object: it is the anti-Wick operator with Weyl symbol ρ. Such operators were first considered by Berezin [2], and have been developed since by many independent authors. A further generalization of (7) now consists in replacing the standard Gaussian φ0 by an arbitrary square integrable function φ and to consider the Weyl transforms of functions of the type  ρ(z) =



μ(z )W φ(z − z )d 2n z = (μ ∗ W φ)(z) .

(8)

The operators thus obtained are called Toeplitz operators (or localization operators) in the mathematical literature; they are natural generalizations of anti-Wick operators [12, 24, 28]. So far, so good. The rub comes from the fact that it is not clear why the Weyl transform ρ

of (7) should indeed be a density operator. For this, ρ

has to satisfy three stringent conditions: (i) ρ

must be positive semidefinite: ρ

≥ 0 and (ii) be self-adjoint: ρ

= ρ

∗ ; (iii) ρ

must be of trace class and have trace one: Tr(

ρ ) = 1. While it is easy to verify (i) (which implies (ii) since L2 (Rn ) is complex), it is the third condition which poses problem since it is certainly not trivially satisfied, as we will see in the course of this paper. Viewed in a broader perspective, quasidistributions of the type ρ = μ ∗ W φ belong to the Cohen class [6, 7, 18, 23], which has a rich internal structure and is being currently very much investigated for its own sake both in time–frequency analysis [3–5, 9, 10, 27] and in quantum mechanics [19].

1.2 Notation and Terminology The scalar product on L2 (Rn ) is defined by  (ψ|φ)L2 =

ψ(x)φ ∗ (x)d n x

(9)

and we thus have (ψ|φ)L2 = φ|ψ in Dirac bra–ket notation; in this notation

φ = |φ φ|. The phase space T ∗ Rn ≡ R2n will be equipped with the canonical Π 3 symplectic structure σ = nj=1 dpj ∧ dxj , given in matrix notation by σ (z, z ) =   0 I J z · z , where J = is the standard symplectic matrix on R2n . We denote by −I 0 x )/h¯ T (z) = T (x, p) = e−i(x p −p

(10)

the Heisenberg–Weyl displacement operator. If a ∈ S  (R2n ) is a symbol on R2n , then the corresponding Weyl operator is

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M. de Gosson

W = OpW (a) = A



aσ (z)T (z)d 2n z,

(11)

where aσ is the symplectic Fourier transform of a, formally defined by aσ (z) =



1 2π h¯

n 

i



e− h¯ σ (z,z ) a(z )d 2n z .

(12)

The Wigner transform of ψ ∈ L2 (Rn ) is defined by W ψ(z) =



1 2π h¯

n  e

i − h py ¯ ψ(x

+ 12 y)ψ ∗ (x − 12 y)d n y .

(13)

Similarly, the cross-Wigner transform of two functions ψ, φ ∈ L2 (Rn ) is defined by W (ψ, φ)(z) =



1 2π h¯

n  e

i − h py ¯ ψ(x

+ 12 y)φ ∗ (x − 12 y)d n y .

(14)

We have  W (ψ, φ)(z)d 2n z = (ψ|φ)L2

(15)

for ψ, φ ∈ L1 (Rn ) ∩ F L1 (Rn ) (F the Fourier transform).

2 Density Operators: Summary A density operator on L2 (Rn ) is by definition a bounded operator ρ

which is semidefinite positive: ρ

≥ 0 (and hence self-adjoint), and has trace Tr(

ρ ) = 1. The set D = D(L2 (Rn )) of density operators is a convex subset of the space L1 = L1 (L2 (Rn )) of trace class operators (recall that the latter is a two-sided ideal of the algebra B = B(L2 (Rn )) of bounded operators on L2 (Rn )). In particular density operators are compact operators, and hence, by the spectral theorem, there exists an orthonormal basis (φj )j of L2 (Rn ) and coefficients 3 3 satisfying λj ≥ 0 and

λ = 1 such that ρ

can be written as a convex sum j j j λj Πφj of orthogonal

φj converging in the strong operator topology, that is, counting the projections Π multiplicities, ρ

ψ =



λj (ψ|φj )L2 φj .

j

By definition, the “Wigner distribution” of ρ

is the function

(16)

Generalized Anti-Wick Quantum States

ρ=



127

λj W φj ∈ L2 (R2n ) ∩ L∞ (R2n )

(17)

j

where W ψj is the usual Wigner transform of φj , defined for ψ ∈ L2 (Rn ) by W ψ(z) =



1 2π h¯

n  e

i − h py ¯ ψ(x

+ 12 y)ψ ∗ (x − 12 y)d n y .

(18)

Proposition 1 The Wigner distribution (17) is (2π h) ¯ −n times the Weyl symbol of ρ

: n ρ

= (2π h) ¯ OpW (ρ).

(19)

The proof of this result immediately follows from:

φ of L2 (Rn ) on Cφ (φ ∈ L2 (Rn )) has Weyl Lemma 1 The orthogonal projection Π n symbol (2π h) ¯ W φ.

φ is the

φ ψ = (ψ|φ)L2 φ so the distributional kernel of Π Proof By definition Π

= OpW (a) function Kφ (x, y) = φ(x)φ ∗ (y); the Weyl symbol of an operator A

by the formula [18, 28] being related to the kernel K of A  a(z) =

e

i − h py ¯ K(x

+ 12 y, x − 12 y)d n y

the lemma follows.

 

A class of windows φ playing a privileged role in the study of density operators is the Feichtinger algebra [14, 16], which is the simplest modulation space (a nonexhaustive list of references on the topic of modulation spaces is [15, 23, 25, 29]. In [18, Chapters 16 and 17] we have given a succinct account of the theory using the Wigner transform instead of the traditional short-time Fourier transforms approach). By definition Feichtinger’s algebra M 1 (Rn ) = S0 (Rn ) consists of all distributions ψ ∈ S  (Rn ) such that W (ψ, φ) ∈ L1 (R2n ) for some window φ ∈ S (Rn ); when this condition holds, we have W (ψ, φ) ∈ L1 (R2n ) for all windows φ ∈ S (Rn ) and the formula  (A1) ||ψ||φ = |W (ψ, φ)(z)|d 2n z = ||W (ψ, φ)||L1 defines a norm on the vector space M 1 (Rn ); another choice of window φ  leads to an equivalent norm and one shows that M 1 (Rn ) is a Banach space for the topology thus defined. We have the following continuous inclusions: S (Rn ) ⊂ M 1 (Rn ) ⊂ C 0 (Rn ) ∩ L1 (Rn ) ∩ F L1 (Rn ) ∩ L2 (Rn )

(A2)

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and S (Rn ) is dense in M 1 (Rn ). Moreover, for every ψ ∈ L2 (Rn ) we have the equivalence ψ ∈ M 1 (Rn ) ⇐⇒ W ψ ∈ L1 (Rn ) .

(A3)

A particularly important property of M 1 (Rn ) is its metaplectic invariance. Let

S ∈ Mp(n) (the metaplectic group) cover S ∈ Sp(n); then W (

Sψ,

Sφ) = W (ψ, φ)◦ S −1 [18]; it follows from this covariance formula and the fact that the choice of φ is irrelevant, that

Sψ ∈ M 1 (Rn ) if and only if ψ ∈ M 1 (Rn ). Also, M 1 (Rn ) is invariant under the action of the translations T (z). One can show [23] that M 1 (Rn ) is the smallest Banach algebra containing S (Rn ) and having metaplectic and translational invariance. It turns out that M 1 (Rn ) is in addition an algebra for both pointwise product and convolution; in fact if ψ, ψ  ∈ M 1 (Rn ), then ||ψ ∗ ψ  ||φ ≤ ||ψ||L1 ||ψ  ||φ so we have L1 (Rn ) ∗ M 1 (Rn ) ⊂ M 1 (Rn ) hence in particular M 1 (Rn ) ∗ M 1 (Rn ) ⊂ M 1 (Rn ) . Taking Fourier transforms we conclude that M 1 (Rn ) is also closed under pointwise product.

3 Toeplitz Operators: Definitions and Properties Let φ (hereafter called window) be in M 1 (Rn ). By definition, the Toeplitz operator

φ = Opφ (a) with window φ and symbol a is A

φ = A



φ (z)d 2n z, a(z)Π

(20)

φ : L2 (Rn ) −→ L2 (Rn ) is the orthogonal projection onto T (z)φ, that is where Π

φ (z)ψ = (ψ|T (z)φ)L2 T (z)φ . Π

(21)

We observe that (ψ|T (z)φ)L2 is, up to a factor, the cross-ambiguity transform of the pair (ψ, φ); in fact [18, § 11.4.1] n (ψ|T (z)φ)L2 = (2π h) ¯ Amb(ψ, φ)(z),

(22)

where Amb(ψ, φ)(z) =



1 2π h¯

n  e

i − h py ¯ ψ(y

+ 12 x)φ(y − 12 x)∗ d n y .

(23)

Generalized Anti-Wick Quantum States

129

φ as We can, therefore, rewrite the definition (20) of A n

φ ψ = (2π h) A ¯



a(z) Amb(ψ, φ)(z)T (z)φd 2n z

(24)

which is essentially the definition of single-windowed Toeplitz operators given in the time–frequency analysis literature (see, e.g., [8–10, 29]). Toeplitz operators are linear continuous operators S (Rn ) −→ S  (Rn ); in view of Schwartz’s kernel theorem they are thus automatically Weyl operators. In fact (cf. Lemma 2.4 of [5] and formula (19) in [8]):

φ = Proposition 2 Let φ ∈ M 1 (Rn ) and a ∈ M ∞ (R2n ). The Toeplitz operator A Opφ (a) has Weyl symbol (2π h) ¯ n (a ∗ W φ), that is n

φ = (2π h) A ¯ OpW (a ∗ W φ) .

(25)

Proof It is sufficient to assume that a ∈ S (R2n ). Let πφ (z) be the Weyl symbol of

φ (z) on T (z)φ; we thus have the orthogonal projection Π

φ (z)ψ|χ )L2 = (Π



πφ (z)W (ψ, χ )(z )d 2n z

for all ψ, χ ∈ S (Rn ) (see, e.g., [18, § 10.1.2]). Lemma 1 and the translational covariance of the Wigner transform [18, § 9.2.2] imply that we have n  n  πφ (z)(z ) = (2π h) ¯ W (T (z)φ)(z ) = (2π h) ¯ W φ(z − z )

and hence n

φ (z)ψ|χ )L2 = (2π h) (Π ¯



W φ(z − z )W (ψ, χ )(z )d 2n z .

(26)

Using definition (20) we thus have, by the Fubini–Tonnelli theorem,

φ ψ|χ )L2 = (A



φ (z)ψ|χ )L2 d 2n z a(z)(Π 

 n = (2π h) ¯

a(z)  

n = (2π h) ¯

 W φ(z − z )W (ψ, χ )(z )d 2n z d 2n z

 a(z)W φ(z − z )d 2n z W (ψ, χ )(z )d 2n z

φ is a = (2π h) hence the Weyl symbol of A ¯ n W φ ∗ μ as claimed.

 

130

M. de Gosson

We next prove an extension of the usual symplectic covariance result [17, 18, 26] S OpW (a)

S −1 OpW (a ◦ S −1 ) =

(27)

for Weyl operators to the Toeplitz case. We recall the associated symplectic covariance formulas [18, § 8.1.3 and 10.3.1]

Sψ)(z) = W (ψ)(S −1 z) . S T (z)

S −1 = T (Sz) , W (

(28)

φ = Opφ (a) and

Corollary 1 Let A S ∈ Mp(n) have projection S ∈ Sp(n). We have

−1 . S Op

Opφ (a ◦ S −1 ) =

S −1 φ (a)S

(29)

Proof We begin by noticing that we have

S −1 φ ∈ M 1 (Rn ) since φ ∈ M 1 (Rn ) −1 hence

S φ is a bona fide window. Since W φ(z − z ) = W (T (z )φ)(z) we have, setting z = S −1 z and using the relations (28), (a ◦ S −1 ) ∗ W φ(z) =

 

=  =  =  =

a(S −1 z )W (T (z )φ)(z)d 2n z a(z )W (T (Sz )φ)(z)d 2n z a(z )W (

S T (z )

S −1 φ)(z)d 2n z a(z )W (

S T (z )

S −1 φ)(z)d 2n z a(z )W (T (z )

S −1 φ)(S −1 z)d 2n z

= (a ∗ W (

S −1 φ)(S −1 z). It follows, using the covariance formula (27) for Weyl operators that 0 0 1 1 S −1 φ)

S OpW (a ∗ W (

OpW (a ◦ S −1 ) ∗ W φ =

S −1 which is precisely (29) in view of formula (25) in Proposition 2.

 

4 Toeplitz Quantum States The following boundedness result was proven by Gröchenig in [22, Thm. 3]; it is a particular case of Thm. 3.1 in Cordero and Gröchenig [8]:

Generalized Anti-Wick Quantum States

131

= OpW (a) is of trace class: A

∈ Proposition 3 Let a ∈ M 1(R2n ), then A

L ≤ C||a||M 1 for some C > 0 (|| · ||L the trace norm L1 (L2 (Rn )) and ||A|| 1 1 and || · ||M 1 a norm on M 1 (R2n )). Our main result makes use of the class of symbols M 1,∞ (R2n ), which is a particular modulation space containing L1 (R2n ). It is defined as follows [23]: a ∈ M 1,∞ (R2n ) if and only if for some (and hence every) b ∈ S (R2n )  sup ζ ∈R2n

|W 2n (a, b)(z, ζ )|d 2n z < ∞ .

(30)

When b describes S (R2n ) the mappings a −→ ||a||b defined by  ||a||b = sup

ζ ∈R2n

|W 2n (a, b)(z, ζ )|d 2n z

(31)

is a family of equivalent norms on M 1,∞ (R2n ) which becomes a Banach space for the topology they define. We have the following important convolution property between the symbol class M 1,∞ (R2n ) and the Feichtinger algebra M 1 (R2n ): M 1,∞ (R2n ) ∗ M 1 (R2n ) ⊂ M 1 (R2n )

(32)

(Prop. 2.4 in [8]). Let us state and prove our main result. We assume that μ is a Borel probability density function on R2n with respect to the Lebesgue measure: μ ∈ L1 (R2n ) and  μ(z) ≥ 0 ,

μ(z)d 2n z = 1 .

(33)

Theorem 1 Let μ ∈ M 1,∞ (R2n ) and φ ∈ M 1 (Rn ) with ||φ||L2 = 1. Then n n ρ

= (2π h) ¯ OpW (μ ∗ W φ) ¯ Opφ (μ) = (2π h)

is a density operator, and there exists C > 0 such that ||

ρ ||L1 ≤ C||μ ∗ W φ||M 1,∞ ||φ||2M 1 . Proof Let us prove that ρ

∈ L1 (L2 (Rn )). In view of Proposition 3 it is sufficient to show that the Weyl symbol a = (2π h) ¯ n (μ ∗ W φ) is in M 1 (R2n ). We begin 1 by observing that the condition φ ∈ M (Rn ) implies that W φ ∈ M 1 (R2n ); this property is in fact a consequence of the more general result Prop.2.5 in [8] but we give here a direct independent proof. Let b ∈ S (R2n ); denoting by W 2n the crossWigner transform on R2n we have (property (15))

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M. de Gosson

!  ! ! ! 2n 2n 2n ! W (W φ, b)(z, ζ )d zd ζ | = |(W φ|b)L2 (R2n ) !! < ∞ ! and hence W φ ∈ M 1 (R2n ) as claimed. In view of the convolution property (32) we

≥ 0 (and hence ρ

∗ = ρ

) have μ ∗ W φ ∈ M 1 (R2n ) as desired. Let us show that ρ for all ψ ∈ L2 (Rn ). We have 

φ (z)ψ|ψ)L2 d 2n z ; (

ρ ψ|ψ)L2 = μ(z)(Π since by definition

φ ψ = (ψ|T (z)φ)L2 T (z)φ Π we get

φ (z)ψ|ψ)L2 = |(ψ|T (z)φ)|2 ≥ 0 . (Π Let us finally prove that Tr(

ρ ) = 1. We have a ∈ M 1 (R2n ) ⊂ L1 (R2n ) hence ([11]; also [18, § 12.3.2]):  −n Tr(

ρ ) = (2π h) a(z)d 2n z = aσ (0) . ¯ But we have 2n aσ = Fσ (W φ ∗ μ) = (2π h) ¯ (Fσ W φ)(Fσ μ)

and hence, since ||φ||L2 = 1,  aσ (0) =

 W φ(z)d 2n z

μ(z)d 2n z = 1

that is Tr(

ρ ) = 1.

 

A typical example is provided by anti-Wick quantization. Choose for window φ the standard coherent state φ0 : φ0 (x) = (π h¯ )−n/4 e−|x|

2 /2h

¯

.

(34)

Its Wigner transform is given by [1, 18, 20, 26] −n −|z| e W φ0 (z) = (π h) ¯

and is thus a classical probability density

2 /h

¯

(35)

Generalized Anti-Wick Quantum States

133

 W φ0 (z)d n z = 1 .

W φ0 (z) > 0 ,

(36)

It follows that μ ∗ φ is itself a probability density and that Tr(

ρ ) = 1 (for an alternative proof, see Boggiatto and Cordero [3, Thm. 2.4]).

5 Discussion and Perspectives In this paper we have sketched the theory of a class of density operators which might be of genuine interest not only in quantum mechanics, but also in time– frequency analysis whose methods are largely based on operator theory methods grandly similar to those used in quantum mechanics (both Sciences are formally very similar, even if they describe different approaches to tangible reality). There are many related questions we have not addressed in this work, and which are open to future research. The most obvious is the study of entanglement and separability properties for Toeplitz operators. Recall that a density operator ρ

on L2 (Rn ) is separable if there exist sequences of density operators ρ

jA on L2 (RnA )) 3 and ρ

jB on L2 (RnB ) (nA + nB = n) and real numbers αj ≥ 0, j αj = 1 such that 

ρ

=

αj ρ

jA ⊗ ρ

jB

j ∈I

where the convergence is for the norm of L1 (L2 (Rn )) (the trace class operators). When ρ

is not separable, it represents the so-called entangled states of quantum mechanics, which play an increasingly important role in both theoretical developments and practical applications (cryptography, quantum computing, etc.). A topic closely related to separability and entanglement questions is that of the determination of the reduced density operator n ρ

A = (2π h) ¯ A OpW (ρA )

where we have set  ρA (zA ) =

RnB

ρ(zA , zB )dzB .

Under which conditions (if any) is ρ

A a Toeplitz operator when ρ

is? Very little is known outside the Gaussian case, and these questions are still largely open at the moment of writing. Any advances in these directions would be very welcome. Acknowledgments This work was written while the author was holding the Giovanni Prodi visiting chair at the Julius-Maximilians-Universität Würzburg during the summer semester 2019. It is my pleasure to thank the referee for useful comments and suggestions.

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References 1. M.J. Bastiaans, Wigner distribution function and its application to first-order optics, J. Opt. Soc. Am. 69, 1710–1716 (1979) 2. F.A. Berezin, Wick and anti-Wick operator symbols, Mathematics of the USSR-Sbornik, 15(4), 577 (1971); Mat. Sb. (N.S.) 86(128), 578–610 (1971) (Russian) 3. P. Boggiatto and E. Cordero, Anti-Wick quantization with symbols in Lp spaces, Proc. Amer. Math. Soc. 130(9), 2679–2685 (2002) 4. P. Boggiatto and E. Cordero, Anti-Wick quantization of tempered distributions. In Progress in Analysis: (In 2 Volumes), pp. 655–662 (2003) 5. P. Boggiatto, E. Cordero, and K. Gröchenig, Generalized anti-Wick operators with symbols in distributional Sobolev spaces, Integr. Equat. Oper. Th. 48(4), 427–442 (2004) 6. L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7, 781–786 (1966) 7. L. Cohen, Time Frequency Analysis, Vol. 778. Prentice hall, 1995 8. E. Cordero and K. Gröchenig, Time-Frequency analysis of localization operators, J. Funct. Anal. 205, 107–131 (2003) 9. E. Cordero and K. Gröchenig, Necessary conditions for Schatten Class Localization Operators, Proc. Amer. Math. Soc. 133(12), 3573–3579 (2005) 10. E. Cordero and L. Rodino, Wick calculus: a time-frequency approach, Osaka J. Math. 42(1), 43–63 (2005) 11. J. Du and M.W. Wong, A trace formula for Weyl transforms, Approx. Theory. Appl. (N.S.) 16(1), 41–45 (2000) 12. M. Engliš, An excursion into Berezin–Toeplitz quantization and related topics, Quantization, PDEs, and Geometry, 69–115, Birkhäuser, Cham, 2016 13. M. Faulhuber, M.A. de Gosson, D. Rottensteiner, Gaussian Distributions and Phase Space Weyl–Heisenberg Frames, Appl. Comput. Harmon. Anal. 48, 374–394 (2020) 14. H.G. Feichtinger, On a new Segal algebra, Monatsh. Math. 92(4), 269–289 (1981) 15. H.G. Feichtinger, Modulation spaces on locally compact abelian groups. Universität Wien. Mathematisches Institut. (1983) 16. H.G. Feichtinger, Modulation spaces: looking back and ahead, Sampling Theory in Signal and Image Processing 5(2), 109 (2006) 17. G.B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989 18. M. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics. Birkhäuser, Basel, 2011 19. M. de Gosson, Introduction to Born–Jordan Quantization: Theory and applications, Springer– Verlag, series Fundamental Theories of Physics, 2016 20. M. de Gosson, The Wigner Transform, World Scientific Publishing Company, 2017 21. M. de Gosson, Quantum harmonic analysis of the density matrix, Quanta 7(1), 74–110 (2018) 22. K. Gröchenig, An uncertainty principle related to the Poisson summation formula, Stud. Math. 1(121), 87–104 (1996) 23. K. Gröchenig, Foundations of time-frequency analysis, Springer Science & Business Media; 2001 24. K. Gröchenig and J. Toft, Isomorphism properties of Toeplitz operators and pseudo-differential operators between modulation spaces, J. Anal. Math. 114(1), 255–283 (2011) 25. M.S. Jakobsen, On a (no longer) new Segal algebra: a review of the Feichtinger algebra, J. Fourier Anal. Appl. 24(6), 1579–1660 (2018) 26. R.G. Littlejohn, The semiclassical evolution of wave packets, Phys. Reps. 138(4–5), 193–291 (1986) 27. F. Luef and E. Skrettingland, Convolutions for localization operators, J. Math. Pures Appl. 118, 288–316 (2018) 28. M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, (1987) [original Russian edition in Nauka, Moskva (1978)] 29. J. Toft, Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces, In: Toft J. (eds) Modern Trends in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol 172. Birkhäuser Basel, 2006

Signal Analysis and Quantum Formalism: Quantizations with No Planck Constant Jean Pierre Gazeau and Célestin Habonimana

Abstract Signal analysis is built upon various resolutions of the identity in signal vector spaces, e.g. Fourier, Gabor, wavelets. Similar resolutions are used as quantizers of functions or distributions, paving the way to a time-frequency or time-scale quantum formalism and revealing interesting or unexpected features. Extensions to classical electromagnetism viewed as a quantum theory for waves and not for photons are mentioned. Keywords Gabor frames · Wavelet frames · Weyl–Heisenberg group · Affine group · Integral quantization · Covariance · Signal processing

1 Introduction: The Planck Constant h ¯ Is for What? Around 1861 Maxwell derived theoretically the equations that bear his name by using a molecular vortex model proposed by Michael Faraday. He was certainly not aware that their content combines the two uppermost physical theories of the next century, namely special relativity and quantum mechanics, even though he was inspired by the relationship among electricity, magnetism, and the speed of light √ (Weber and Kohlrausch, 1855), c = 1/ "0 μ0 . To see that, let us Fourier transform the second-order Maxwell equations in the vacuum 

∂2 ∂2 ∂2 ∂2 − − − c2 ∂t 2 ∂x 2 ∂y 2 ∂z2

 ψ(t, r) = 0 ,

r = (x, y, z) ,

(1)

J. P. Gazeau () Astroparticules et Cosmologie, Université Paris Diderot, Paris, France e-mail: [email protected] C. Habonimana Université du Burundi and École Normale Supérieure, Bujumbura, Burundi © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_8

135

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where ψ stands generically for components of magnetic or electric fields or potentials. We obtain the following equation understood in the sense of distributions,  

(ω, k) = 0 , ω2 − c2 k2 ψ

(ω, k) = ψ

1 4π 2

 R4

e−i(ωt− k· x) ψ(t, r) dt d3 x ,

(2) where ω is the angular frequency (in radians per second), and k = (kx , ky , kz ), k2 := k · k, is the wave vector (in radians per metre). One can read (2) as the quadratic relation between multiplication frequency and wave vector operators acting in Fourier representation,  

(ω, k) = 0 , Ω 2 − c2 K2 ψ

(ω, k) = (ω, k) ψ

(ω, k) , (Ω , K) ψ

(3)

and, equivalently, in space-time coordinates, Ω ψ(t, r) = −i

∂ ψ(t, r) , ∂t

K ψ(t, r) = −i∇ r ψ(t, r) .

(4)

Hence, we can see in these apparently trivial manipulations two “modern” aspects of the so-called classical electromagnetism formulated by Maxwell: 1. the underlying quantization of frequency and wave vector, ω → Ω, k → K, 2. the underlying invariant equation of relativistic dynamics, ω2 − c2 k2 = 0. The second one can be viewed as a particular case of the Einstein energy-momentum equation E 2 − c2 p2 = m2 c4 with m = 0 (massless particle) after insertion of one proportionality constant allowing to write E ∝ ω, p ∝ k. This constant should have the dimension [ML2 T−1 ] of an action. It is precisely the constant h¯ introduced in 1900 by Planck to explain properly the electromagnetic radiation emitted by a black body. The relations E = h¯ ω, p = hk, ¯ were subsequently proposed by de Broglie in his 1924 thesis to account for wave-particle duality. Hence, we can discern the role of the Planck constant in bridging two worlds, namely classical electromagnetism with phase space {(time-space, frequency-wave vector) ∈ R8 } (no rest mass here) and quantum mechanics built from the classical phase space {(space, momentum) ∈ R6 } , and the heuristic quantization rules r → R and p → P (respectively, multiplication and derivation operators in space representation). These rules yield the well-known wave equations like Schrödinger, Klein–Gordon, Dirac. . . . However, there is a trouble concerning the pair (time-energy) due to the lack of existence of a consistent time operator in quantum mechanics (see, for instance, the review [1]): according to Pauli’s argument [2], there is no self-adjoint time operator canonically conjugating

Classical or Quantum?

137

to a Hamiltonian if the Hamiltonian spectrum is bounded from below. Here we have displayed the quantum nature of Maxwell equations, which look like the Klein– Gordon equation with no h¯ . Restoring the latter as a global factor would appear as purely artificial since there is precisely no mass! Note that at the meeting World Metrology Day held on 20 May 2019 it was definitely decided to anchor the S.I. standard of mass, the kilogram, to the Planck constant whose value is henceforth fixed to h = 2π h¯ = 6.62607015 × 10−34 kg m2 s−1 . Elementary solutions of the Maxwell equations are those Fourier exponentials (basic wave planes) used to implement the Fourier transform in (2). We will see that there is no sound argument preventing the existence of a time operator. Thus, what is really the status of frequency, time, wave vector, . . . , in electromagnetism? Classical observables? Quantum observables? Now, waves in classical electromagnetism are signals, in the true sense of the latter. This is the leitmotiv of the present contribution, in which we revisit signal analysis where physical quantities are just time and frequency, or time and scale, as they are illustrated by the two phase spaces in Fig. 1. The organisation of this contribution is as follows. Section 2 is a brief overview of the basic methods in Signal Analysis, namely Fourier, Gabor, and Wavelet. In Sect. 3 we explain the relationship between signal analysis and quantum formalism by pointing out their common Hilbertian framework and the existence in both cases of the essential resolution of the identity. We then define what we mean by quantization and semi-classical portrait together with their probabilistic content and a possible classical limit, both resulting from a given resolution of the identity. In Sect. 4 we implement our approach to quantizations with projector-valued measures provided by Fourier or Dirac bases. They are trivially equivalent to the respective spectral decompositions of the self-adjoint time and frequency operators, but not allow quantizations of functions of time and frequency. We enter the heart of our aims and results in Sects. 5 and 6 with resolutions of the identity resulting from the Weyl–Heisenberg and affine groups, and weight functions on the plane

Fig. 1 Left: Time-frequency plane ∼ Weyl–Heisenberg (Classical observable f (t, ω)). Right: Time-scale half-plane ∼ Affine group (Classical observable f (t, a))

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or half-plane, respectively. The first one stands for the translational symmetry of the time-frequency plane, while the second one stands for the translationdilation symmetry of the time-scale half-plane (Fig. 1). Each one has a unitary irreducible representation whose square integrability allows to establish resolution of the identities through Schur’s Lemma and implement the corresponding covariant integral quantizations. We insist on the fact that whatever the choice of the weight function, time and frequency or scale operators remain essentially the same. In Sect. 7 we illustrate the previous material with examples of operators acting on signals and built from functions f (b, ω) through Gabor quantization. This gives an idea of the wide range of possibilities in signal analysis offered by our procedure. In Sect. 8 we discuss some aspects of our results which we consider as open questions. Our contribution may appear as somewhat speculative. We hope that it will open the way to new directions not only in Signal Analysis but also in Physics through the unveiling of some quantum features of the so-called classical physics. All the results are given without proof. The latter can be found in previous works [3–9].

2 Fourier, Gabor, and Wavelet Analysis in a Nutshell In this section we recall the basics of these three types of signal analysis and fix our Hilbertian notations in terms of Dirac ket(s), |· , as vectors in a Hilbert space and bra(s), ·|, as elements of its dual. A temporal “finite-energy” signal s(t) is viewed as a vector denoted by |s , or by the abusive |s(t) , in the Hilbert space L2 (R, dt). Its energy is precisely s 2 = s|s .

2.1 Fourier Analysis, Like in Euclidean Geometry. . . Fourier analysis rests upon the family of elementary signals √1 eiωt , ω ∈ R. 2π These non-square-integrable functions can be considered as forming a “continuous orthonormal basis” in the following sense 5

! 6 ∞ ! 1 1 1  iωt ! iω t = ei(ω −ω)t dt = δ(ω − ω) √ e ! √ e 2π 2π 2π

“orthonormality” .

−∞

1=

! 65 ∞ ! ! 1 ! ! √ eiωt √1 eiωt ! dω ! 2π ! 2π

“continuous basis solving the identity” .

−∞

Then the inverse Fourier transform is viewed as the Hilbertian decomposition in elementary signals :

Classical or Quantum?

139

! ! 1 √ eiωt !! s(t) 2π 7 89 :

! 6 ! 1 ! √ eiωt dω , ! 2π

5

∞ |s(t) = −∞

Fourier transform sˆ (ω)= √1

 +∞

2π

−∞

e−iωt s(t)dt

together with the norm or energy conservation (Plancherel) ∞

s =

∞ |s(t)| dt =

2

|ˆs (ω)|2 dω = ˆs 2 .

2

−∞

−∞

2.2 Gabor Signal Analysis (∼ Time-Frequency) The ingredients of the Gabor transform, or time-frequency representation, of a signal are translation combined with modulation. One chooses a probe, or window or Gaboret, ψ which is well localised in time and frequency at once, and which is normalised, ψ = 1. This probe is then translated in time and frequency, but its size is not modified (in modulus): ψ(t) → ψb,ω (t) = eiωt ψ(t − b) .

(5)

The time-frequency or Gabor transform is then :  s(t) → S[s](b, ω) ≡ S(b, ω) = ψb,ω |s =

+∞ −∞

e−iωt ψ(t − b) s(t) dt.

It is easy to prove that there is conservation of the energy : 

s 2 =

+∞ −∞

 |s(t)|2 dt =

+∞  +∞

−∞

−∞

|S(b, ω)|2

db dω def = S 2 , 2π

and so the reciprocity or reconstruction formula holds as: +∞  +∞

 s(t) =

−∞

−∞

S(b, ω)eiωt ψ(t − b)

db dω . 2π

This reconstruction holds in the Hilbertian sense. It results from resolution of the identity provided by the continuous non-orthogonal family {ψb,ω , (b, ω) ∈ R2 } (overcompleteness):  1=

+∞  +∞

−∞

−∞

! ;< db dω !! ψb,ω ψb,ω ! . 2π

(6)

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2.3 Continuous Wavelet Transform (∼ Time-Scale) In this analysis, one also picks a well localised “mother wavelet” or probe ψ(t) ∈ L2 (R, dt), but we impose more on it: its Fourier transform should be zero at the ∞

(0) = √1 origin, which implies zero average for ψ, i.e. ψ ψ(t) dt = 0, and 2π

with modulus even. Such conditions are encapsulated in: ∞ 0 < cψ := 2π

dω ˆ = |ψ(ω)| ω

∞

2

0

2 ˆ |ψ(−ω)|

−∞

dω < ∞. ω

0

Then we build the (continuous) family of translated-dilated-contracted versions of ψ: 1 ψb,a (t) = √ ψ a



t −b a

2 . a∈R&+ ,b∈R

We obtain an overcomplete family in L2 (R, dt), which means that any signal decomposes as 1 |s(t) = cψ

∞

∞ db

−∞

! 6  ! 1 da t −b ! . S(b, a) ! √ ψ a a2 a

(7)

0

The coefficient S(b, a), as a function of the two continuous variables b (time) and a (scale), is the continuous wavelet transform of the signal: 5

1 S(b, a) = √ ψ a



!   ∞ 1 t −b t − b !! s(t) dt . √ ψ ! s = a a a

(8)

−∞

These equations derive from the resolution of the identity provided by the nonorthogonal |ψba ’s: 1 1= cψ

∞

∞ db

−∞

da a2

! 6 5 !   ! 1 1 t − b !! !√ ψ t − b √ ψ !. ! a a a a

(9)

0

Energy conservation holds as well, but its repartition in the half-plane stands with respect to the Lobachevskian geometry determined by the measure db da/a 2 . The latter is left-invariant under the affine transformations (b, a) → (b , a  )(b, a) = (b + a  b, a  a).

Classical or Quantum?

1

s = cψ

141

∞

2

∞ db

−∞

da |S(b, a)|2 , a2

∞ cψ ≡ 2π

0

2 ˆ |ψ(ω)|

0

dω . |ω|

3 From Signal Analysis to Quantum Formalism 3.1 Resolution of the Identity as the Common Guideline Given a measure space (X, μ) and a (separable) Hilbert space H , an operatorvalued function X x → M(x) acting in H , resolves the identity operator 1 in H with respect to the measure μ if  M(x) dμ(x) = 1

(10)

X

holds in a weak sense. In Signal Analysis, analysis and reconstruction are grounded in the application of (10) on a signal, i.e. a vector in H H |s

reconstruction

=

 9 analysis :7 8 M(x)|s dμ(x) . X

In quantum formalism, integral quantization is grounded in the linear map of a function on X to an operator in H  f (x) →

f (x) M(x) dμ(x) = Af ,

1 → 1 .

X

3.2 Probabilistic Content of Integral Quantization: Semi-Classical Portraits If the operators M(x) in  M(x) dμ(x) = 1 X

(11)

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J. P. Gazeau and C. Habonimana

are nonnegative, i.e. φ|M(x)|φ ≥ 0 for all x ∈ X, one says that they form a (normalised) positive operator-valued measure (POVM) on X. If they are further unit trace-class, i.e. tr(M(x)) = 1 for all x ∈ X, i.e. if the M(x)’s are density operators, then the map f (x) → fˇ(x) := tr(M(x)Af ) =



f (x  ) tr(M(x)M(x  )) dμ(x  ) X

is a local averaging of the original f (x) (which can very singular, like a Dirac!) with respect to the probability distribution on X, x  → tr(M(x)M(x  )) . This averaging, or semi-classical portrait of the operator Af , is in general a regularisation, depending of course on the topological nature of the measure space (X, μ) and the functional properties of the M(x)’s.

3.3 Classical Limit Consider a set of parameters κ and corresponding families of POVM Mκ (x) solving the identity  X

Mκ (x) dμ(x) = 1 .

(12)

One says that the classical limit f (x) holds at κ 0 if fˇκ (x) :=

 X

f (x  ) tr(Mκ (x)Mκ (x  )) dμ(x  ) → f (x)

as

κ → κ0 ,

where the convergence fˇ → f is defined in the sense of a certain topology. Otherwise said, tr(Mκ (x)Mκ (x  )) tends to tr(Mκ (x)Mκ (x  )) → δx (x  ), where δx is a Dirac measure with respect to μ, 

f (x  ) δx (x  ) dμ(x  ) = f (x) .

X

Of course, these definitions should be given a rigorous mathematical sense, and nothing guarantees the existence of such a limit.

Classical or Quantum?

143

4 First Examples of Operators M(x): Projector-Valued (PV) Measures for Signal Analysis The measure space is (R, dx), and the Hilbert space is L2 (R, dx). Variable x represents time, x = t, or represents frequency, x = ω. The identity is solved by two types of continuous “orthogonal bases” 1. Dirac basis  +∞|δt ≡ |t for time analysis (trivial sampling), based on the wellknown −∞ δ(t − t  ) f (t  ) dt  = f (t), {δt , t ∈ R} ,

δt |δt  = δ(t − t  ) ,



+∞ −∞

|δt δt | dt = 1 ,

with resulting analysis-reconstruction  s(t) =

+∞

−∞

δ(t − t  ) s(t  ) dt  ⇔ |s =



+∞ −∞

|δt δt |s dt ,

δt |s = s(t).

√ 2. Fourier basis χω (t) = eiωt / 2π for frequency analysis {χω , ω ∈ R} ,

χω |χω = δ(ω − ω ) ,



+∞ −∞

|χω χω | dω = 1 ,

with resulting Fourier analysis-reconstruction  s(t) =

+∞ −∞

 χω (t)ˆs (ω) dω ⇔ |s =

+∞ −∞

|χω χω |s dω ,

χω |s = sˆ (ω).

In the next we apply the Fourier PV quantization to the elementary time and frequency variables.

4.1 PV Measures for Quantization: Time Operator The time operator T ≡ At is obtained as  t → At = T =

+∞

−∞

t|δt δt | dt .

From |s  := T |s =



+∞

−∞

t|δt δt |s dt , 7 89 : s(t)

δt |s  = ts(t) ,

(13)

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J. P. Gazeau and C. Habonimana

one sees that T is the multiplication operator (T s)(t) = ts(t) This operator, with domain the Schwartz space S , is essentially self-adjoint in L2 (R, dx) and (13) is nothing but its spectral decomposition.

4.2 PV Measures for Quantization: Frequency Operator In turn, the frequency operator Ω is  ω → Aω ≡ Ω =

+∞ −∞

ω |χω χω |s dω .

(14)

From 



|s := Ω|s =

δt |s  =



+∞ −∞

+∞

−∞

ω|χω χω |s dω , 7 89 : sˆ (ω)

eiωt



ω sˆ (ω) √ dω = −i ∂t 2π

+∞ −∞

eiωt sˆ (ω) √ dω , 2π

it is the derivation operator (Ωs)(t) = −i ∂t s(t). Symmetrically to the previous case, this operator, with domain the Schwartz space S , is essentially self-adjoint in L2 (R, dx) and (13) is nothing but its spectral decomposition.

4.3 Mutatis Mutandis. . . and CCR In a symmetrical way we can write  Ω=

−∞

 T =

+∞

+∞

−∞

ω|δω δω | dω

(Ω sˆ )(ω) = ω sˆ (ω) ,

t |χt χt | dt ,

(T sˆ )(ω) = i∂ω sˆ (ω) .

Time and frequency operators obey the “canonical” commutation rule (a CCR with no h¯ !) T Ω − Ω T ≡ [T , Ω] = i I , with its immediate Fourier uncertainty consequence

Classical or Quantum?

145

Δs T Δs Ω ≥

1 , 2

Δs A :=

'

s|A2 |s − (s|A|s )2 .

Now, one should keep in mind that CCR [A, B] = i1 for a self-adjoint (A, B) pair, with common domain, holds true only if both have continuous spectrum (−∞, +∞). The expression of the CCR in terms of the respective unitary operators reads as eiσ Ω eiτ T = eiσ τ eiτ T eiσ Ω ,

(Weyl relations) .

(15)

von Neumann proved (1931) [10, 11] that up to multiplicity and unitary equivalence the Weyl relations have only one solution (see [12] for the proof).

4.4 Limitations of PV Quantization The previous quantizations based on spectral PV measures have clearly very limited scopes. As a matter of fact, they are just equivalent to the spectral decomposition of functions of t or of functions of ω:  f (t) → Af = fˆ(ω) → Af =

+∞

−∞



+∞

−∞

dt f (t) dω fˆ(ω)

multiplication operator on L2 (R, dt) |δt δt | |χt χt | (pseudo-)differential operator on L2 (R, dω) multiplication operator on L2 (R, dω) |δω δω | |χω χω | (pseudo-)differential operator on L2 (R, dt).

So, the question is how to manage time-frequency functions f (t, ω)?

5 Weyl–Heisenberg Covariant Integral Quantization with Gabor and Beyond 5.1 From PV Quantization to Gabor POVM Quantization In order to manage time-frequency functions f (b, ω), we naturally think to the resolution of the identity provided by the Gabor POVM introduced in (6) that we remind below.  db dω 1= |ψbω ψbω | , ψbω |ψb ω = δ(b − b ) δ(ω − ω ), 2 2π R unit-norm probewhere δt |ψbω = eiωt ψ(t − b) are the modulated-transported  vectors in L2 (R, dt) and where R ⊃ Δ → Δ db2πdω |ψbω ψbω | is the correspond-

146

J. P. Gazeau and C. Habonimana

ing normalised positive operator-valued measure on the plane. Then the quantization of f (b, ω) is given by  f → Af =

R2

db dω f (b, ω) |ψbω ψbω | . 2π

(16)

The corresponding semi-classical portrait is given by fˇ(b, ω) =

 R2

db dω f (b , ω ) |ψbω |ψb ω |2 . 2π

(17)

Applied to the time and frequency variables, we find that nothing basic is lost with regard to the Fourier PV quantization: Ab = T + Cst1 1 ,

(18)

Aω = Ω + Cst2 1 ,

(19)

where the additive real constants are easily cancelled through an appropriate choice of ψ.

5.2 Beyond Gabor Quantization Gabor signal analysis and quantization are the simplest ones among a world of possibilities, all of them being based on the unitary dual of the Weyl–Heisenberg group. Let us remind the most important features of this group that we use in our approach to quantization. More details are given in the pedagogical [6]. We recognise in the construction of the Gabor family (5) the combined actions of the two unitary operators introduced in (15), with respective generators the selfadjoint time and frequency operators   L2 (R, dt) ψ(t) → ψb,ω (t) = eiωT e−ibΩ ψ (t) .

(20)

Two alternative forms of the action (20) are provided by the Weyl formulae (15) combined with the Baker–Campbell–Hausdorff formula:     bω ψb,ω (t) = eibω e−ibΩ eiωT ψ (t) = ei 2 ei(ωT −bΩ) ψ (t) .

(21)

In the above appears the Weyl or displacement operator up to a phase factor ei(ωT −bΩ) = D G (b, ω) ,

ψb,ω (t) = ei

bω 2

  D G (b, ω)ψ (t) .

(22)

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The appearance of this phase factor like that one appearing in the composition formula (15) indicates that the map (b, ω) → D G (b, ω) is a projective representation of the time-frequency abelian plane. Dealing with a true representation necessitates the introduction of a third degree of freedom to account for this phase factor. Hence we are led to work with the Weyl–Heisenberg group GWH . GWH = {g = (ς, b, ω) , ς ∈ R , (b, ω) ∈ R2 } ,

(23)

with neutral element: (0, 0, 0), and   1 g1 g2 = ς1 + ς2 + (ω1 b2 − ω2 b1 ) , b1 + b2 , ω1 + ω2 , g −1 =(−ς, −b, −ω) . 2 (24) The Weyl–Heisenberg group symmetry underlying the Gabor transform is understood through its unitary irreducible representation (UIR). As a result of the von Neumann uniqueness theorem, any infinite-dimensional UIR, U , of GWH is characterised by a real number λ = 0 (there is also the degenerate, one-dimensional, UIR corresponding to λ = 0). If the Hilbert space carrying the UIR is the space of finite-energy signals H = L2 (R, dt), the representation operators are defined by the action similar to (20) (with the choice λ = 1) and completed with a phase factor: U (ς, b, ω) = eiς e−iωb/2 eiωT e−ibΩ = eiς D G (b, ω) .

(25)

With this material in hand, it is easy to prove the Weyl–Heisenberg covariance of the Gabor transform. S[U (0, b0 , ω0 )s](b, ω) = ψb,ω |U (0, b0 , ω0 )s   ωb , b, ω ψ|s = U (0, −b0 , −ω0 ) U 2  ! 5  ! ωb 1 − (ω0 b − ωb0 ), b − b0 , ω − ω0 ψ !! s = U 2 2 = ei(ω−ω0 /2)b0 S[s](b − b0 , ω − ω0 ) .

(26)

We now pick a bounded trace-class operator Q0 on H . Its unitary Weyl–Heisenberg transport yields the continuous family of bounded trace-class operators Q(b, ω) = U (ς, b, ω)Q0 U (ς, b, ω)† = D G (b, ω)Q0 D G (b, ω)† . Applying the Schur Lemma to the irreducible projective unitary representation (b, ω) → D G (b, ω) allows to prove the resolution of the identity obeyed by the operator-valued function Q(b, ω) on the time-frequency plane

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 R2

Q(b, ω)

db dω = 1. 2π

It ensues the Weyl–Heisenberg covariant integral quantization in its most general formulation:  db dω . (27) f (b, ω) → Af = f (b, ω) Q(b, ω) 2 2π R The Gabor quantization corresponds to the choice Q0 = |ψ ψ|. There exists an equivalent form of the quantization (27) which is expressed in terms of D G , the Weyl transform of the operator Q0 defined as the “apodization” function on the time-frequency plane Π (b, ω) := Tr (U (−b, −ω)Q0 ) ,

(28)

and the symplectic Fourier transform of f (b, ω),  Fs [f ](b, ω) :=



R2



e−i(bω −b ω) f (b , ω )

One obtains  db dω Af = , U (b, ω) Fs [f ](b, ω) Π (b, ω) 2 2π R

db bω . 2π

(29)

Π (b, ω) = Tr (U (−b, −ω)Q0 ) ,

with Fs [f ](b, ω) := Fs [f ](−b, −ω). The semi-classical portrait of Af reads 

. / db dω  (b − b, ω − ω) f (b , ω ) ΠΠ 2π R2  . /  dt  dω  (b − b, ω − ω) f (b , ω ) = Fs [Π ] ∗ Fs Π , 4π 2 R2

fˇ(b, ω) =

 ω) = Π (−b, −ω). With a true probabilistic content, the meaning of where Π(b, the convolution . /  Fs [Π ] ∗ Fs Π is clear: it is the probability distribution for the difference of two vectors in the timefrequency plane, viewed as independent random variables, and thus is adapted to the abelian and homogeneous structure of the latter (choice of origin is arbitrary!). In a certain sense the function Π corresponds to the Cohen “f ” function [13] (for more details see [14] and the references therein) or to Agarwal–Wolf filter functions [15]. The simplest choice is Π (b, ω) = 1, of course. Then Q0 = 2P, where P is the parity operator. This no filtering choice yields the popular Weyl–Wigner integral

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quantization equivalent to the standard (∼ canonical) quantization. Another example sin bω , which presents appealing aspects is the Born–Jordan weight, Π (b, ω) = bω [16].

6 Affine Quantization Like the Weyl–Heisenberg symmetry underlies the properties of the Gabor transform and quantization, the affine symmetry of the open upper half-plane Π+ := {(b, a) | p ∈ R , a > 0} explains the properties of the continuous wavelet transform (8) and the resulting quantization. Together with 1. the multiplication law 

b2 (b1 , a1 )(b2 , a2 ) = b1 + , a1 a2 a1

 ,

2. the unity (0, 1), 3. and the inverse −1

(b, a)

  1 , = −ab, a

Π+ is viewed as the affine group Aff+ (R) of the real line (b,a)

R x → (b, a) · x = b +

x . a

Note that we adopt here a definition for the dilation which is the inverse of the standard one used in Sect. 2.3. It conveniently allows to get rid of the factor 1/a 2 present in the Lobachevskian measure in (7) and (9). With the above definitions, the measure on Aff+ (R) which is left-invariant with respect to its internal law reads da db, i.e. is canonical. The affine group Aff+ (R) has two non-equivalent UIR U± . Both are squareintegrable and this is the rationale behind continuous wavelet analysis resulting from a resolution of the identity (9). U± and the UR U = U+ ⊕ U− are realised in the Hilbert space H = L2 (R, dt) = H+ ⊕ H− , where H± are the (Hardy) subspaces of finite-energy signals with positive and negative frequencies, respectively. Here we restrict our choice to U+ , which is more conveniently realised in the present context through the Fourier transform of signals with positive frequencies ω ≡ x > 0. Its action is defined as eibx  x  . L2 (R∗+ , dx) φ(x) → U+ (b, a)φ(x) = √ φ a a

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Following the approach developed in [5], we consider the operator -

 *

M

:=

R×R∗+

−1 C−1 DM U+ (b, a)CDM * (b, a) db da ,

CDM φ(x) :=

2π φ(x) , x (30)

with the set of assumptions: 1. The weight function * (b, a) is C ∞ on Π+ . 2. It defines a tempered distribution with respect to the variable b for all a > 0. 3. The operator M* is self-adjoint bounded on L2 (R∗+ , dx). With these assumptions, the action of M* on φ in L2 (R∗+ , dx) is given in the form of the linear integral operator 

∞

(M* φ)(x) =

M * (x, x  ) φ(x  ) dx  .

(31)

0

Its kernel M * is given by  x 1 x −x, . M * (x, x  ) = √ *

p x 2π x 

(32)

In the above *

p is the partial Fourier transform of * with respect to the variable b: 1 *

p (y, a) = √ 2π



+∞ −∞

e−iby * (b, a) db .

(33)

From Schur’s Lemma, one easily proves that the affine transport of M* resolves the identity:  1=

R×R∗+

db da * M (b, a) , cM*

M* (b, a) := U+ (b, a)M* U+† (b, a) ,

(34)

√  +∞ da provided that 0 < cM* := 2π 0

b (1, −a) < ∞. The particular case a * (9) holding for the continuous wavelet transform with probe ψ corresponds to the weight function whose partial Fourier transform is *

p (y, a) =

 y √ 1

(−y) ψ

− , 2π ψ a a

a > 0,y < 0.

(35)

It results from (34) the affine covariant integral quantization of a function or distribution on the half-plane:  f (b, a) → A* f =

R×R∗+

db da f (b, a) M* (b, a) . cM*

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This map is covariant with respect to the unitary affine action U+ : † * U+ (b0 , a0 )A* f U+ (b0 , a0 ) = AU(b

0 ,a0 )f

,

(36)

with     a , (U(b0 , a0 )f ) (b, a) = f (b0 , a0 )−1 (b, a) = f a0 (b − b0 ), a0

(37)

U being the left regular representation of the affine group when f ∈ L2 (Π+ , db da). The action of A* on φ in C0∞ (R∗+ ) is given in the form of the linear integral operator  (A* f φ)(x) =

0

+∞

Af* (x, x  ) φ(x  ) dx  ,

(38)

with kernel Af* (x, x  )

=

1 cM*

x x

 0

+∞

   x ˆ dq x  *

p −q,  fp x − x, . q x q

(39)

For the quantization of the variables b and a (almost) nothing basic is lost: * (A* b φ)(x) = −i∂x φ(x) + Cst3 , and (Aa φ)(x) = Cst4 x φ(x). With an appropriate choice of the weight function * one gets the values Cst3 = 0 and Cst4 = 1, and * * * so [A* a , Ab ] = i1. Since Aa is bounded below, Ab , although symmetric, is not self-adjoint and has no self-adjoint extension.

7 Examples of Operatorial Signal Analysis Through Gabor Quantization We illustrate the content of this contribution with examples showing the link between standard operatorial tools of Signal analysis, like filtering, multiplication, convolution, etc., and Gabor quantization (16). For a given choice of the probe ψ the latter maps a function (or tempered distribution) f (b, ω) on the time-frequency plane to the integral operator Af in the Hilbert space of finite-energy signals defined by  (Af s)(t) = δt |Af |s = with integral kernel given by

+∞

−∞

dt  Af (t, t  ) s(t  ) ,

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1 Af (t, t ) = √ 2π 



+∞

−∞

db f ω (b, t  − t) ψ(t − b) ψ(t  − b).

Here f ω (b, y) is the partial Fourier transform with respect to the variable ω: 1 f ω (b, y) = √ 2π



+∞

−∞

dω f (b, ω) e−iωy .

Let us go through some specific situations. The Gabor quantization of separable functions f (b, ω) = u(b)v(ω) yields the integral kernel v(t ˆ  − t) √ 2π

Auv (t, t  ) =



+∞ −∞

db u(b) ψ(t − b) ψ(t  − b) ,

and so the action on a signal s(t) reads as the combination of convolution and multiplication 1 (Auv s)(t) = √ 2π



  db ψ(b) u(t − b) ψ˜ b vˆ˜ ∗ s (t) ,

+∞ −∞

ψb (t) := ψ(t − b) .

Therefore, in the monovariable case f (b, ω) = u(b) one gets the multiplication operator 





Au (t, t ) = δ(t − t)

+∞

−∞

  db u(b) |ψ(t − b)|2 = δ(t  − t) |ψ|2 ∗ u (t)

  (Au(b) s)(t) = |ψ|2 ∗ u (t) s(t) . For the other monovariable case f (b, ω) = v(ω) one gets the integral kernel v(t ˆ  − t) Av(ω) (t, t ) = √ 2π 



+∞ −∞

˜ − t  − b) = db ψ(b) ψ(t

v(t ˆ  − t) Rψψ (t − t  ), √ 2π (40)

where  Rψψ (t) :=

+∞

−∞

  ψψ (t) dt  ψ(t  ) ψ(t  − t) = ψ ∗ ψ˜ (t) = R

is the autocorrelation of the probe, i.e. the correlation of the probe with a delayed copy of itself as a function of delay. Note that 0 1 1 ˆ 2 (t) . √ Rψψ (t) = F −1 |ψ| 2π We eventually get the convolution operator on the signal:

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 1 1 0 Rψψ vˆ˜ ∗ s (t). (Av(ω) s)(t) = √ 2π It is amusing to explore the above quantization formulae when the signal itself or its Fourier transform is quantized. Precisely, given a normalised probe ψ and a signal s(t), what is the operator As , i.e. the Gabor quantization of the signal itself? It is given by     As(b) s (t) = |ψ|2 ∗ s (t) s(t) . This means a kind a self-control of the original signal with its regularised version |2 . In turn, its Fourier yielded by the convolution with the probability distribution |ψ transform sˆ (ω), what is the operator Asˆ ? Applying (40) the Gabor quantization of the Fourier transform sˆ (ω) of the signal yields the convolution operator:  / 1 . Rψψ s ∗ s (t) . (Asˆ s) (t) = √ 2π It is a kind of an autocorrelation of the original signal with one of its regularised versions yielded by a superposition of multiplication operators. It is actually the sˆ (ω) which yields the autocorrelation of the signal weighted Gabor quantization of  by the autocorrelation of the probe  +∞   1 A s (t) = √ dt  Rψψ (t  ) s(t  ) s(t  − t) . sˆ 2π −∞ It is equally inspiring to Gabor quantize the Gabor transform of the signal s  S(b, ω) =

+∞ −∞

e−iωt ψ(t − b) s(t) dt .

One obtains the (involved) convolution:  (AS s) (t) =

+∞

−∞

db ψ(t − b)

.   / ψb s ∗ ψ−b s (t) ,

ψb (t) = ψ(t − b) .

It is worthy to examine all these formulae with the most immediate probe choice, namely the normalised centred Gaussian with width σ t2 − 1 ψ(t) ≡ Gσ (t) = 1/4 √ e 2σ 2 . π σ Its autocorrelation is also a (not normalised) Gaussian

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RGσ Gσ (t) = e

−

t2 4σ 2

(41)

.

The integral kernel of the quantization of f (b, ω) reads



Af (t, t ) = √

=√

(t − t  )2  4σ 2 e −

1 2π σ 1 2π σ

+∞

−∞

e

−

(t

− t  )2 4σ 2

(b − (t + t  )/2)2 σ2 db f ω (b, t − t) e 

−

⎛

⎞ (·)2   − ⎜

 2 ⎟ t +t σ . ⎝fω (·, t − t) ∗ e ⎠ 2

Finally, note that the resulting semi-classical portrait of the operator Af is the double Gaussian convolution: fˇ(b, ω) =

 R2

− db dω f (b , ω ) e 2π

(b − b )2 σ 2 (ω − ω )2 − 2 2σ 2 . e

As a consequence we observe that no classical limit holds at σ → 0 or σ → ∞. This is just an illustration of the time-frequency uncertainty principle.

8 Discussion As was illustrated in the above section, it can be profitable to view any linear operator used in signal processing, e.g. convolution, “quantization”, compression, etc., as the quantum version of some classical f (b, ω) or f (b, a). A tentatively complete conversion table is being established (doctoral program of C. Habonimana). New tools of signal analysis can be established in this way. Now, in the quantum framework derived from Hilbertian signal analysis, measured (set of data!) finite-energy signal s(t) (resp. sˆ (ω)) becomes “quantum states” or “wave functions” and can be given a probabilistic interpretation of some significance, e.g. localisation measurement in time (resp. frequency or scale). Hence, for a given signal s, in some experiment or trial, for some f (t, ω) or f (t, a), how to interpret the “expected values” s|Af |s , e.g. s|T |s (resp. s|Ω|s )? This mean time (resp. mean frequency) represents a kind of characteristic time (resp. frequency) for a phenomenon (e.g. an earthquake, a sound, . . . ) encoded into s. It is tempting to consider the class of (deterministic or not) signals as eigenstates of some operator Af resulting from the integral quantization of a function or distribution f (t, ω), like Ωeiωt = ωeiωt . . . .

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The last but not the least point, one is not familiar with considering classical electromagnetic fields or waves as quantum states, on which act operators which are built from functions or distributions f (x, k) on the eight-dimensional phase space time - space × frequency - wave vector along the quantization procedure exposed here. Many features of quantum formalism, like entanglement or quantum measurement, remain to be explored within this original framework, where the absence of the Planck constant offers the opportunity to work with no limitation of scale. Acknowledgments J.-P. G. is indebted to the École Doctorale de l’Université du Burundi (UB) and his Director, Prof. Juma Shabani, for hospitality and financial support. He is also indebted to the Centre International de Mathématiques Pures et Appliquées for financial support.

References 1. J.G. Muga, R. Sala Mayato, I.L. Egusquiza, (Eds.), Time in Quantum Mechanics, Lecture Notes in Physics Monographs, Springer-Verlag Berlin Heidelberg (2008). 2. J. W. Pauli, in Encyclopedia of physics, edited by S. Flugge (Springer, Berlin, 1958), Vol. 5, p. 60. 3. H. Bergeron and J.-P. Gazeau, Integral quantizations with two basic examples, Ann. Phys. 344, 43 (2014). 4. H. Bergeron, E.M.F. Curado, J.-P. Gazeau, and Ligia M.C.S. Rodrigues, Weyl-Heisenberg integral quantization(s): a compendium, arXiv:1703.08443, new version in progress 5. J.-P. Gazeau, R. Murenzi Covariant affine integral quantization(s), J. Math. Phys. 57, 052102 (2016). arXiv:1512.08274 6. J.-P. Gazeau, From classical to quantum models: the regularising rôle of integrals, symmetry and probabilities, Found. Phys. 48 1648–1667 (2018); arXiv:1801.02604. 7. H. Bergeron and J.-P. Gazeau Variations à la Fourier-Weyl-Wigner on quantizations of the plane and the half-plane, Entropy 20 787-1-16 (2018). 8. H. Bergeron, E. Czuchry, and J.-P. Gazeau, and P. Małkiewicz Integrable Toda system as a novel approximation to the anisotropy of Mixmaster, Phys. Rev. D 98 083512 (2018). 9. J.-P. Gazeau, T. Koide, and D. Noguera Quantum Smooth Boundary Forces from Constrained Geometries, J. Phys. A: Math. Theor 52 445203 (2019); arXiv:1902.07305v3 [quant-ph] 10. J. von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), 570–578. 11. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Berlin, Springer, 1932. 12. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. 13. L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1966) 781–786. 14. L. Cohen, The Weyl operator and its generalization, Springer Science & Business Media, 2012 15. B.S. Agarwal and E. Wolf, Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics, Phys. Rev. D 2 (1970) 2161; 2187; 2206. 16. M. de Gosson, Born-Jordan Quantization: Theory and Applications, Springer 2016.

Quantitative Methods in Ocular Fundus Imaging: Analysis of Retinal Microvasculature Demetrio Labate, Basanta R. Pahari, Sabrine Hoteit, and Mariachiara Mecati

Abstract Several diseases including diabetes, hypertension, and glaucoma are known to cause alterations in the human retina that can be visualized noninvasively and in vivo using well-established techniques of fundus photography. Since the treatment of these diseases can be significantly improved with early detection, methods for the quantitative analysis of fundus imaging have been the subject of extensive studies. Following major advances in image processing and machine learning during the last decade, a remarkable progress is being made towards developing automated quantitative methods to identify image-based biomarkers of different pathologies. In this paper, we focus especially on the automated analysis of alterations of retinal microvasculature—a class of structural alterations that is particularly important for early detection of cardiovascular and neurological diseases. Keywords Retina imaging · Representation learning · Segmentation

1 Introduction The retina is a layered tissue covering the interior part of the eye that is responsible for image formation. Since its function requires it to receive direct light from the outside world, the retina is uniquely accessible for imaging noninvasively and in vivo. In addition, the retinal tissue is metabolically active and, as an extension of the central nervous system, it exhibits a close affinity to the brain tissue in terms of anatomy, functionality, and response to insult [2, 63, 79]. It follows that not only diseases of the eye but also circulatory and neurological diseases can D. Labate () · B. R. Pahari · S. Hoteit Department of Mathematics, University of Houston, Houston, TX, USA e-mail: [email protected]; [email protected]; [email protected] M. Mecati Dipartimento di Automatica e Informatica, Politecnico di Torino, Torino, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_9

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manifest in the retina and several studies have demonstrated that retinal alterations may be predictive of a range of diseases of the eye (e.g., diabetic retinopathy, glaucoma) as well as other important systemic diseases (e.g., hypertension, stroke, and Alzheimer’s disease), making retinal imaging a major subject of investigation. Due to its importance for monitoring the health status of the eye, retinal imaging has a long history and photographic techniques were already applied at the end of the nineteenth century [42]. Retinal fundus photography, in particular, was originally introduced by Gullstrand in 1910 [45] to visually record the rear of the eye, also known as the fundus. As successive studies have established the importance of retinal imaging to detect not only disease of the eye but also a range of systemic diseases, fundus photography has developed very rapidly. Due to its safety, cost effectiveness and non-invasive nature, this imaging tool has become a mainstay of the eye clinical care and it is now widely employed for large scale, population-based screening. Despite the emergence of new imaging modalities, the technology of fundus photography is still evolving and a new generation of inexpensive, portable, easy-to-operate fundus cameras is revolutionizing retinal screening programs and becoming widely available [80]. Evaluation of fundus images for medical diagnostics requires expert clinicians whose time is valuable and whose judgment is necessarily subjective. As a consequence, reliable automated approaches to retinal image analysis are in great need not only to improve diagnostic accuracy but also to increase clinician productivity in routine population screening settings. In response to this need, multiple methods have been proposed and implemented with focus on the segmentation of blood vessels in retinal fundus images, quantification of their alterations, and detection of abnormalities. Since assessing the type and degree of such alterations or abnormalities is critical to detect the presence of a disease or discriminate among different diseases, targeted image analysis methods have been designed to reconstruct the vessel structure of the retina and extract anatomic landmarks including the macula, the fovea, and the optic disc [2, 81]. While the literature on retinal image analysis is extensive (over 750 papers published to date) and includes a number of excellent reviews [2, 14, 81], in this paper, we focus specifically on the analysis of retinal microvascularization and computerized methods designed to quantify corresponding morphological changes in fundus images. Alterations in retinal microvascularization are known to correlate not only with eye disease (e.g., diabetic retinopathy) but also with cardiovascular and brain diseases since changes in retinal microvasculature may reflect similar changes occurring in cerebral microvasculature [16]. To reliably detect and accurately classify such vascular changes, a number of algorithms have been proposed that apply a variety of methods from statistics, classical and fractal geometry as well as neural networks. The goal of this paper is to survey conventional and the state-of-the-art methods in this active area of investigation with special focus on emerging ideas from multiscale representations and deep learning. The rest of the paper is organized as follows. In Sect. 2 we present some background material on the anatomy on the retina and fundus photography. In Sect. 3 we review the existing literature on the analysis of retinal microvascularization and

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illustrate the application of emerging techniques to this task including most notably methods from deep learning and advanced multiscale representations.

2 Anatomy of the Retina and Retinal Imaging We briefly review the anatomy of the retina and how retinal imaging is useful to detect and monitor a range of diseases.

2.1 Anatomic Structure of the Retina We start with some notes about the anatomy of the retina and refer to [32, 59, 78] for additional details. The retina is a thin (about 0.5 mm thick) transparent tissue lining the inner surface of the posterior region of the eye. It is composed of multiple layers of specialized sensory neurons that are interconnected through synapses. Light that enters the eye is captured by photoreceptor cells (the so-called rod and cones cells) located in the outermost layer of the retina and then converted into an electrical signal that eventually reaches the retinal ganglion cells. The axons of these cells form the optic nerve and, through this nerve, electrical signals are relayed to the higher visual processing centers that enable us to perceive visual images. The central region of the retina (see Fig. 1) contains an oval-shaped pigmented region called macula that is responsible for the fine, high resolution vision. This region is thicker than peripheral retina due to the higher density of photoreceptors, especially cones, and their associated bipolar and ganglion cells as compared with Fig. 1 Anatomy of the eye. The eye is our organ of sight and it consists of several components including the cornea, iris, pupil, lens, sclera, retina, macula, fovea, choroid, and optic nerve. The retina is a tissue lining the inner surface of the posterior region of the eye

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peripheral retina. The fovea, located near the center of the macula, is a small pit containing the largest concentration of cone cells. As for any other tissue, the retina receives blood through the vascular system and this system is visible from the pupil, as in ophthalmoscopy or retinal photography. There are two sources of blood supply to the retina: the central retinal artery and the choroidal blood vessels. The choroidal blood vessels carry most of the blood flow and are responsible for the maintenance of the outer retina and the photoreceptors; the rest of the blood flows to the retina through the central retinal artery from the optic nerve head to nourish the inner retinal layers. The central retinal artery has four main branches that are clearly visible in retinal photography (see Fig. 2) where the vessels emerging from the optic nerve head are displaced in a radial fashion curving toward and around the fovea.

2.2 Retinal Manifestations of Disease Many diseases manifest themselves in the retina [2, 65] including not only diseases of the eye, such as glaucoma, age-related macular degeneration, and diabetic retinopathy, but also cardiovascular diseases, such as hypertension and atherosclerosis, and brain diseases such as Alzheimer’s disease. In fact, the retinal vascular network is optimized for efficient flow and alterations from this state that are observed on fundus images may be indicative of vascular damage and manifestation of disease processes. Such alterations may include: changes in morphological characteristics of the blood vessels, e.g., changes in thickness and tortuosity; blood vessel abnormalities, e.g., hemorrhages, microaneurysms, neovascularizations; abnormalities of the pigment epithelium, e.g., drusen; choroidal abnormalities, e.g., choroidal lesions. Fig. 2 Retinal circulation visualized through fluorescein angiography. Fluorescein angiography is a medical procedure in which a fluorescent contrast dye is injected into the bloodstream. The dye highlights the blood vessels located in the back of the eye (the ocular fundus) so that they can be visualized and photographed

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Critical problems are how to detect saddle changes that may be associated with early stages of a disease and how to discriminate alterations associated with different diseases.

2.3 Fundus Photography As indicated above, fundus photography has been a seminal tool for the early developments in diagnoses of eye conditions and is still the most widely used in clinical practice. It requires a fundus camera consisting of an optical system comprising a specialized low power microscope with an attached camera capable of simultaneously illuminating and imaging the rear of the eye; this includes the retina, posterior pole, optic disc, and macula [43]. Initially designed as a film-based imaging system, more recently, with the emergence of digital imaging, the use of digital cameras in fundus photography has allowed to achieve more flexibility in image manipulation, faster processing, and easier transmission of information. During the last decade, the technology of fundus photography has further developed and a new generation of inexpensive, portable, easy-to-operate fundus cameras has become widely available [80]. One main limitation of fundus photography is that it generates a 2-D representation of the 3-D retinal tissue projected onto the imaging plane and, thus, it is not very effective to resolve the vessels located in the deeper retinal layers. Optical coherent tomography (OCT), introduced in the 1990s [39], uses the principle of low-coherence interferometry to generate 3-D reconstructions of the retina and is rapidly becoming a standard. Even though we do not consider OCT images in this paper, most methods of image analysis discussed in this paper apply to OCT data with minor changes.

2.4 Image Dataset To facilitate the development fundus image analysis methods and provide benchmark against which to compare the performance of different algorithms, a number of research groups have created publicly available, annotated fundus image databases. The most notable are the following. • MESSIDOR database [71]. It includes 1200 color fundus images taken with resolutions ranging from 1440 × 960 pixels to 2304 × 1536 pixels [51]. Each image is categorized into one of four groups, corresponding to a diabetic patients without diabetic retinopathy and three increasing stages of diabetic retinopathy. • DRIVE database [33]. It was established to enable comparative studies on segmentation of retinal blood vessels in fundus images. It contains 40 fundus images of size 768 × 584 pixels from subjects with diabetes, of which seven

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Fig. 3 Representative fundus image (right) from the DRIVE image database and corresponding manual segmentation (right) also available in the database

show signs of diabetic retinopathy and the rest appears healthy. The set is divided into a training and a test set, both containing 20 images. For the training images, a manual segmentation of the vasculature is available (see Fig. 3). For the test cases, two manual segmentations are available; one given as gold standard, the other one given to compare computer generated segmentations with those of an independent human observer. • STARE database [93]. This database contains 402 retinal fundus images annotated by domain experts for 44 possible manifestations (features) visible in each image. Manual segmentation of the vasculature is also included for a subset of 40 images of size 605 × 700. • HRF database [50]. It contains 45 color fundus images divided into 15 images of healthy patients, 15 images of patients with diabetic retinopathy and 15 images of patients with glaucoma. These images were acquired using a Canon CR-1 fundus camera with a field of view of 60◦ and have a size of 3504 × 2336 pixels. The database also includes a manual segmentation of the vessel network performed by human experts.

3 Automated Image Analysis of Retinal Vascularization Characteristics of the retinal vascular structure are potential indicators of various diseases and the first attempts to automatically extract quantitative information that could be relevant for clinical diagnostics can be traced back to the pioneering work of Matsui et al. [69]. Spurred by the emergence of digital retinal photography and digital filter-based image analysis techniques, retinal image processing developed dramatically in the 1990s. These developments resulted in a large number of publications focusing on the digital reconstruction of retinal vessel and quantification

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of local vessel characteristics such as vessel width and branching angles or global ones such as the fractal dimension of the whole vessel network. Many such studies, though, especially before the year 2000, were mostly descriptive; while they did establish an association between local or global changes of retinal vascularization and a disease, they could not predict the presence of a disease based on observed vascular changes. Only during the last 10–15 years, with the emergence of methods from machine learning, an increasing number of publications did focus on applying measures of retinal vascularization to disease prediction and biomarkers discovery. Together with such advances, studies during the last decade have established that alterations in retinal vascularization are not only associated with eye diseases but also with cardiovascular and brain diseases since changes in retinal microvasculature may reflect similar changes occurring in blood flow dynamics and cerebral microvasculature. Since retinal vessels are the only segment of the human microcirculation that can be observed directly, these discoveries have further motivated the effort in developing computerized methods of retinal image analysis capable of predicting the insurgence of cardiovascular diseases such as hypertension [27, 101] or cerebral disease such as AD [16, 29, 48]. Below, we survey classical and the state-of-the-art results in this active area of investigation.

3.1 Local Measures of Retinal Vascularization Several classical studies have focused on extracting local measurements from retinal fundus images. To generate objective measures so that images from different patients can be compared, the effect of image magnification due to the photographic acquisition has to be removed either by taking the magnification into account or by defining dimensionless measurements. We list below local measurements most commonly found in the literature. Retinal vessel width or caliber. Several algorithms have proposed methods to extract retinal vessel width automatically or semi-automatically [20, 40]. A number of papers have investigated the relevance of retinal vascular caliber and its variability in connection with diabetic retinopathy and hypertension [21, 56, 74, 85, 96]. However, fundus images are subject to magnification and affected by potential refractive error so that measurements recorded from any particular individual cannot be directly compared with another individual. Hence, alternative dimensionless measurements have been proposed, most notably the arteriolar-venular ratio (AVR) and the length-diameter ratio. Arteriolar-venular ratio. It is defined as the ratio between the average diameters of the arterioles with respect to the venules and was first proposed by Stokoe and Turner [94] in connection with the study of vascular changes in patients undergoing treatment for hypertension. This quantity has been further investigated for its potential association with cardiovascular disease [58, 103].

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Length-diameter ratio. It is calculated as the length from the midpoint of a particular vascular bifurcation to the midpoint of the preceding bifurcation, expressed as a ratio to the diameter of the parent vessel at the bifurcation. This is another dimensionless measure that is affected by changes in vascular caliber and has been found to be increased in hypertension [55]. Angles at vessel bifurcations. Another measurable parameter of blood vessel topography is the angle subtended between two branching offspring blood vessels at a bifurcation junction. Studies have shown that this angle is reduced in hypertension [92]. Junctional exponents. It is defined as the X exponent in the mathematical relationship D0X = D1X + D2X , where D0 is the diameter of the parent vessel, D1 and D2 are the diameters of the offspring vessels. This expression is based on the physical observation that arterial diameters at any bifurcation should conform to a relationship that minimizes shear stress in a vascular network [105]. The junctional exponent has been calculated to be approximately equal to X = 3 when vascular network costs are minimized and this value has been confirmed experimentally [106]. Studies have shown that retinal junctional exponents deviate from optimal values with advancing age and hypertension [92], and in association with vascular disease [19]. Vessel tortuosity. This parameter is the subject of multiple studies and several definitions are proposed in the literature [1, 83]. Even though there is no unique definition, it is often defined as the ratio between the length of a vessel from A to B and the shortest distance between points A and B drawn by a straight line. The degree of vascular tortuosity has been associated with a number of vascular and nonvascular diseases such as diabetic retinopathy, cerebrovascular disease, stroke, and ischemic heart disease [25, 26, 88] and has been used as a measure of disease severity in retinopathy of prematurity [25, 100]. Neovascolarization. It describes the sprouting of new vessels from pre-existent ones and has been associated with aging and eye disease as age-related macular degeneration [18].

3.2 Global Measures of Retinal Vascularization Since individual local measures of retinal vascularization do not convey sufficient information to capture the complexity of the retinal vascular branching pattern and since many diseases have been observed to be associated with a general microvascular remodeling of the retina, several studies have proposed methods to quantify the overall geometric complexity of retinal vasculature. Fractal analysis. A large number of papers have proposed the use of fractal geometry to measure the pattern characteristics of the retinal vascular branches; in a Google Scholar search, we found 174 publications whose titles contain the words “retinal” and “fractal.” Central to such methods is the concept of fractal

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dimension D that provides a quantitative measure of the degree of complexity of a geometric set as a ratio of the change in detail to the change in scale [66]. For a classical rectifiable curve f , we have that D(f ) = 1 (matching its topological dimension) while, for more irregular curves, D(f ) > 1 and D(f ) = 2 if a curve is plane-filling curve. Family, Masters, and Platt [35, 67] were the first to introduce fractal analysis to quantify retinal vascular branching patterns and found that, in normal retina, D(f )  1.7 in accord with the theoretical diffusion limited aggregation model by Witten and Sander [102]. Following this work, other studies observed that larger fractal dimension of the retinal vasculature, reflecting higher geometric complexity of the retinal vascular branching pattern, is associated with early signs of retinopathy [22] and that lower fractal dimension, reflecting reduction in the retinal vasculature complexity, is observed with aging [9, 10]. Without attempting to list all contributions to this topic, we recall that changes in retinal vascular fractal dimension have been shown also to be related to hypertension [24, 28, 109], stroke [9, 23, 31, 57], and mortality from coronary heart disease [61]. However, the measurement of the fractal dimension of retinal vessels is very sensitive to image quality and the method used for its computation [53]. In addition, studies have shown that the fractal dimension alone is unable to differentiate vasculatures with very similar fractal dimension but different structures in their fluid dynamic design and function [53, 107] and that additional information is needed to assess the pathological vascular states [68]. To overcome this limitation, researchers in the field have proposed more sophisticated tools from fractal geometry including the notion of Fourier Fractal dimension [9] and the application multifractal analysis where the fractal dimension D is replaced by a measure consisting of a vector providing a more accurate description of the underlying geometry [95, 97, 98]. Yet another recent approach combines fractal analysis with graph-based method [6] where first a graph is extracted from the retina blood vessel structure with the nodes representing branching or end points and the edges representing vessel segments; fractal analysis is then used to characterize the extracted graphs. Orientation measures. With the emergence of advanced multiscale methods about 2005, a number of researchers proposed methods to analyze and quantify retinal vessel networks based on multiscale directional representations. Among such contributions, we recall in particular the work by Bekkers et al. [13, 49] that applies a sophisticated geometric approach based on theory of best exponential curve fits in the roto-translation group to define a new tortuosity measure. We also recall the work by Sing et al. [90] that applies a system of multiscale directional filters inspired by the theory of shearlets to generate a measure of vessel organization called orientation score. Even though the orientation score was originally applied for the analysis of neuronal images, the same method can be applied directly on retinal images [51]. We report in Table 1 the application of these measures to the analysis of fundus images on the MESSIDOR dataset. The table shows that, as compared with the fractal dimension, these alternative

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Table 1 Average value of tortuosity measure [13], orientation score [90], fractal dimension [53] on the MESSIDOR dataset Subgroup R0 R1 R2 R3

Mean ± STD Tortuosity measure [13] 1.624 ± 0.120 1.657 ± 0.124 1.698 ± 0.122 1.795 ± 0.160

Orientation score [90] 0.0866 ± 0.0166 0.0885 ± 0.0156 0.0894 ± 0.0160 0.0930 ± 0.0163

Fractal dimension [53] 1.3864 ± 0.0324 1.3852 ± 0.0345 1.3781 ± 0.0364 1.3869 ± 0.0384

Data consists of four subgroups, R0–R3, where R0 are the healthy cases and R1–R3 are diabetic retinopathy cases in increasing degree of severity. Representative images from the MESSIDIOR dataset for the four subgroups are shown in Fig. 4

Fig. 4 Left panel: Representative images from MESSIDOR dataset. Starting from top left, clockwise: healthy retina, low-degree retinopathy, medium-degree retinopathy, high-degree retinopathy. Right panel: corresponding segmented images obtained from the B-Cosfire algorithm [11]

geometric measures provide a more insightful description of the changes in retinal vasculature associated with diabetic retinopathy. Machine learning methods. During the last decade, several research teams have applied methods from machine learning for the automated detection of retinal diseases on fundus images taken from databases or retinal screening programs [8, 17]. The basic idea in such studies consists in extracting features from retinal images and then apply supervised classifiers such as support vector machines (SVM), random forests or naive Bayes classifiers to separate images into distinct pathological classes, such as healthy and diabetic retinopathy cases. Typically these studies extract a multiplicity of retinal image features that include both vascular features, such as vessel width and fractal dimension, and nonvascular ones, such as lesions, exudates, and hemorrhages (cf. [4, 7, 82, 86]). More recently, following the spectacular success of deep learning algorithms in many classification tasks, an increasing number of studies is applying deep learning algorithms, especially Convolutional Neural Networks (CNN), to predict the presence of a disease in fundus images. One major difference with respect to more traditional machine learning methods is that deep learning algorithms learn features directly from the raw images avoiding the use of hand-

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Table 2 Diabetic retinopathy classification performance—measured in AUC—on the MESSIDOR dataset using different classification methods Publication Antal and Hajdu [7] Gargeya and Leng [41] Giancardo et al. [44] Giancardo et al. [44] Roychowdhury et al. [86]

Features Lesions + anatomy Deep features Vasculature embedding Vasculature embedding + microaneurysms Lesions + anatomy

Classifier Ensemble CNN SVM SVM Hierarchical

AUC 0.989 0.940 0.678 0.865 0.904

Images are classified into two classes as healthy vs diabetic retinopathy

designed or model-based features [89]. While these algorithms typically require a relatively large number of training (labeled) samples to learn a satisfactory model of the disease class, they are more flexible than conventional machine learning methods and—when properly trained—can perform very competitively. However, the need of many training samples can be a serious limitation in the clinical application of this approach [30]. Similar to classification using traditional machine learning, also the features learned by the deep learning algorithms are usually not limited to vascular features. In most cases, even though the algorithm is not designed to explicitly detect lesions (e.g., hemorrhages, microaneurysms), it implicitly learns to recognize them when it extracts local features [41, 46]. In some applications, the deep learning approach is explicitly designed to detect lesions [3]. However, one can apply a deep learning framework to efficiently learn features of retinal vascularization by using a representation learning approach, as recently proposed by Giancardo et al. [44]. Their approach generates a vasculature embedding by leveraging the internal representation of a specially designed CNN trained endto-end with the raw pixels and manually segmented vessels. Table 2 compares the classification performance of multiple machine learning and deep learning algorithm on the MESSIDOR dataset where images are assigned either to the healthy or to the diabetic retinopathy class. Performance metric is the area under the curve (AUC) obtained by measuring the area under the receiver operating characteristic (ROC) curves that is created by plotting the true positive rate against the false positive rate at various threshold settings.

3.3 Segmentation of Retinal Vessels Automated segmentation of the vascular network in fundus images is a nontrivial task due to the variable size of the vessels, the relative low intensity contrast, and the possible occurrence of abnormalities such as hemorrhages and microaneurysms [15]. Over the last approximately 20 years, this topic has been the subject of a large number of studies with over a hundred publications including several excellent

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Table 3 Retinal blood vessels segmentation performance—measured in Sensitivity (Se), Specificity (Sp), AUC and Dice similarity coefficient (DSC)—on the DRIVE dataset using different supervised and unsupervised algorithms Publication Unsupervised Azzopardi et al. [11] Fraz et al. [36] Miri and Mahloojifar [72] Roychowdhury et al. [87] Zhao et al. [108] Supervised Liskowski and Krawiec [62] Lupascu et al. [64] Oliveira et al. [76] Orlando et al. [77] Soares et al. [91] Vega et al. [99]

Se

Sp

AUC

DSC

0.7655 0.7152 0.7352 0.7249 0.7420

0.9704 0.9768 0.9795 0.9830 0.9820

0.9614 – 0.9458 0.9620 –

– 0.7642 – – –

0.7520 0.6728 0.8039 0.7897 0.7283 0.7444

0.9806 0.9874 0.9804 0.9684 0.9788 0.9600

0.9710 0.9561 0.9821 0.9506 0.9614 –

– – – 0.7857 – 0.6884

reviews [5, 38, 75]. In general, existing algorithms for the segmentation of the vascular network in fundus images can be divided into two groups. The first one consists of unsupervised or rule-based methods that include morphological operators [36, 70], adaptive thresholding [54], variational methods [87, 108], vesseltracking [12, 104], multiscale, and/or orientable filters [11, 34, 47, 60, 72]. The second group consists of supervised methods (which require manually labeled images for training) and includes classical methods of machine learning and pattern recognition [37, 64, 84, 91] as well as several methods appeared during the last few years based on convolutional neural networks [52, 62, 76, 77, 99]. To date retinal vessel segmentation in fundus images is a generally well understood problem [73]. The field did benefit enormously from the availability of annotated images in several publicly available databases that made it possible to develop and compare the performance of various algorithms. Table 3 compares the segmentation performance of multiple traditional and the state-of-the art strategies evaluated on DRIVE. Performance is assessed using the standard metrics of Sensitivity (Se) or Recall, Specificity (Sp) and Dice similarity coefficient (DSC) or F1 score defined below. By comparing the algorithm pixel classification with respect to the gold standard labeling (manual segmentation), we determine the number of true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN). Then we obtain Se =

TP , T P + FN

Sp =

TN , T N + FP

DSC =

2T P . 2T P + F P + F N

Sensitivity measures the ability of the method to properly detect blood vessels, while specificity measures its capability of distinguishing the other non-vessel structures.

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DSC is a measure of the overall performance of the algorithm. It achieves its maximum value of 1 when the segmentation is perfect and its lowest value of 0 when the segmentation is completely wrong. Acknowledgments M.C. acknowledges the hospitality of the Department of Mathematics at the University of Houston where this work was initiated. D.L. acknowledges partial support of NSF DMS 1720487 and 1720452.

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A Time–Frequency Analysis Perspective on Feynman Path Integrals S. Ivan Trapasso

Abstract The purpose of this expository paper is to highlight the starring role of techniques from time–frequency analysis in some recent contributions concerning the mathematical theory of Feynman path integrals. We hope to draw the interest of mathematicians working in time–frequency analysis on this topic, as well as to illustrate the benefits of this fruitful interplay for people working on path integrals. Keywords Path integral · Modulation spaces · Pseudodifferential operators

1 Introduction The path integral formulation of non-relativistic quantum mechanics is a paramount contribution by Richard Feynman (Nobel Prize in Physics, 1965) to modern theoretical physics. The origin of this approach goes back to Feynman’s Ph.D. thesis of 1942 at Princeton University (recently reprinted, cf. [8]) but was first published in the form of research paper in 1948 [23]; see also [62] for some historical hints. In rough terms we could say that this approach provides a quantum counterpart to Lagrangian mechanics, while the standard framework for canonical quantization as developed by Dirac relies on the Hamiltonian formulation of classical mechanics. Path integrals and Feynman’s deep physical intuition were the main ingredients of the celebrated diagrams, introduced in the 1949 paper [24], which gave a whole new outlook on quantum field theory. For a first-hand pedagogical introduction we recommend the textbook [25], where it is clarified how the physical intuition of path integrals comes from a deep understanding of the lesson given by the two-slit experiment. We briefly outline below the main features of Feynman’s approach. Recall that the state of a nonrelativistic particle in the Euclidean space Rd at time t is represented by the wave S. I. Trapasso () Dipartimento di Scienze Matematiche (DISMA) “G. L. Lagrange”, Politecnico di Torino, Torino, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 P. Boggiatto et al. (eds.), Landscapes of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-3-030-56005-8_10

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function ψ(t, x), (t, x) ∈ R × Rd , such that ψ(t, ·) ∈ L2 (Rd ). The time-evolution of a state ϕ(x) at t = 0 is governed by the Cauchy problem for the Schrödinger equation:  i h∂ ¯ t ψ = (H0 + V (x))ψ ψ(0, x) = ϕ(x),

(1)

where 0 < h¯ ≤ 1 is a parameter (the Planck constant), H0 = −h¯ 2 ./2 is the standard Hamiltonian for a free particle, and V is a real-valued potential; we conveniently set m = 1 for the mass of the particle. The map U (t, s) : ψ(s, ·) → ψ(t, ·), t, s ∈ R, is a unitary operator on L2 (Rd ) and is known as propagator1 or evolution operator; we set U (t) for U (t, 0). Since U (t) is a linear operator we can formally represent it as an integral operator, namely  ψ(t, x) =

Rd

ut (x, y)ϕ(y)dy,

where the kernel ut (x, y) (we also write ut,s (x, y) or u(t, s)(x, y) for the kernel of U (t, s)) is interpreted as the transition amplitude from the position y at time 0 to the position x at time t. In a nutshell, Feynman’s prescription is a recipe for this kernel, the main ingredients being all the possible paths from y to x that the particle could follow. The contribution of each interfering alternative path to the total probability amplitude is a phase factor involving the action functional evaluated on the path, that is  t L(γ (τ ), γ˙ (τ ))dτ, S [γ ] = S(t, 0, x, y) = 0

where L is the Lagrangian functional of the underlying classical system. Therefore, the kernel should be formally represented as  ut (x, y) =

i

e h¯ S[γ ] Dγ ,

(2)

that is a sort of integral over the infinite-dimensional space of paths satisfying the conditions above. This intriguing picture is further reinforced by the following remark: a formal application of the stationary phase method shows that the semiclassical limit h¯ → 0 selects the classical trajectory, hence we recover the principle of stationary action of classical mechanics.

1 We remark that in physics literature the term “propagator” is usually reserved to the integral kernel

ut of U (t), see below. This may possibly lead to confusion since it is in conflict with the traditional nomenclature adopted in the analysis of PDEs.

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It is well known after Cameron [9] that Dγ cannot be a Lebesgue-type measure on the space of paths, neither it can be constructed as a Wiener measure with complex variance—it would have infinite total variation. The literature concerning the problem of putting formula (2) on firm mathematical ground is huge; the interested reader could benefit from the monographs [4, 29, 50] as points of departure. We will describe below only two of the several schemes which have been manufactured in order to give a rigorous meaning to (2), the type of techniques involved ranging from geometric to stochastic analysis; these approaches both rely on operator-theoretic strategies and are called sequential approach and time-slicing approach. Basically, one is lead to study sequences of operators on L2 (Rd ) which converge to the exact propagator U (t) in a sense to be specified, the strength of convergence competing against the regularity of the potential V . This is the point where time–frequency analysis enters the scene. Techniques of phase space analysis are indeed very well suited to the study of path integrals, the reasons being manifold. First of all, pseudodifferential and Fourier integral operators can be effectively treated from a time–frequency analysis perspective as evidenced by a now vast literature—we highlight [11, 12, 31, 32] among others. Typical function spaces for this purpose are modulation and Wiener amalgam spaces, which may serve as space of symbols as well as background where to investigate boundedness and related properties (algebras for composition, sparsity, diagonalization, etc.). In the same spirit, many results are known on dispersive nonlinear PDEs, in particular on the Schrödinger equation. Notably, function spaces of time–frequency analysis enjoy a fruitful balance between nice properties (Banach spaces/algebras, embeddings, decomposition, etc.) and regularity of their members. The purpose of this overview is to concisely witness some results of this successful interplay, which has made possible to advance in the quest for a rigorous theory of path integral with remarkable results. In particular, we are going to describe three recent contributions on the topic: 1. convergence of time-slicing approximations in Lp spaces (with loss of derivatives) for p = 2—based on [54]; 2. convergence of non-smooth time-slicing approximations inspired by the custom in physics and chemistry—based on [56]; 3. pointwise convergence of the integral kernels of the sequential approximations— based on [57]. First we provide a concise exposition of the two operator-theoretic approaches to path integrals mentioned above. We also collect some preliminary concepts in a separate section for the sake of clarity. Notation We denote by S (Rd ) the Schwartz space of rapidly decaying smooth functions on Rd and by S  (Rd ) the space of temperate distributions. We set x = (1 + |x|)1/2 , x ∈ Rd . The space of smooth bounded functions on Rd with bounded 0 in microlocal derivatives of any order is denoted by Cb∞ (Rd ) (also known as S0,0 analysis); it is equipped with the family of seminorms

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4 4

f k = sup 4∂ α f 4L∞ < ∞, |α|≤k

k ∈ N0 = 0, 1, 2, . . ..

The conjugate exponent p of p ∈ [1, ∞] is defined by 1/p + 1/p = 1. We write f  g if the underlying inequality holds up to a constant factor C > 0, that is f ≤ Cg.

2 A Few Facts on Modulation Spaces In this section we set the function space framework for the rest of the paper. The reader is urged to consult [30, 66–68] for more details and the proofs of the mentioned properties. Modulation spaces were introduced by Feichtinger in the 1980s [21, 22]. At first, they can be thought of as Besov spaces with cubic geometry, namely characterized by isometric boxes in the frequency domain instead of dyadic annuli. To be precise, fix an integer d ≥ 1; for any 1 ≤ p, q ≤ ∞ and s ∈ R we set p,q

Ms

(Rd ) :=

⎧ ⎪ ⎨ ⎪ ⎩

⎛ f ∈ S  (Rd ) : f M p,q = ⎝



⎞1/q k qs k f Lp ⎠

k∈Zd

q

⎫ ⎪ ⎬ d and p ≥ 2 [53, Thm. 8].

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3.2 The Time-Slicing Approximation We now consider another scheme that could be informally called “the Japanese way” to rigorous path integrals, since the leading players in its construction were Fujiwara and Kumano-go, with further developments by Ichinose and Tsuchida. The main references for this approach are the papers [27, 28, 36, 37, 43–45] and the monograph [29], to which the reader is referred for further details. Let us briefly reconsider Eq. (5) and its interpretation in terms of finitedimensional approximations along broken lines; a similar result can be achieved without recourse to the Trotter formula as detailed below. First, let us specify the class of potentials involved in this approach. Assumption (A) The potential V : R × Rd → R satisfies ∂xα V ∈ C 0 (R × Rd ) for any α ∈ Nd0 and |∂xα V (t, x)| ≤ Cα ,

|α| ≥ 2,

(t, x) ∈ R × Rd

for suitable constants Cα > 0. For this wide class of smooth, time-dependent, at most quadratic potentials Fujiwara showed [27, 28] that the propagator U (t, s), 0 < s < t, is an oscillatory integral operator (for short, OIO) of the form 1 U (t, s)ϕ(x) = (2π i h(t ¯ − s))d/2

 Rd

i

e h¯ S(t,s,x,y) a(h, ¯ t, s)(x, y)ϕ(y)dy,

(7)

for some amplitude function a(h, ¯ t, s) ∈ Cb∞ (R2d ). In concrete situations, except for a few cases, there is no hope to compute the exact propagator in an explicit, closed form. Due to this difficulty and inspired by the free particle operator (4), one is lead to consider approximate propagators (parametrices), such as E

(0)

1 (t, s)ϕ(x) = (2π i h(t ¯ − s))d/2

 Rd

i

e h¯ S(t,s,x,y) ϕ(y)dy.

(8)

In view of the previous remarks, this operator is supposed to provide a good approximation of the U (t, s) for t − s small enough. The case of a long interval [s, t] can be treated by means of composition of such operators in the spirit of the time-slicing method proposed by Feynman: given a subdivision Ω = t0 , . . . , tL of the interval [s, t] such that s = t0 < t1 < . . . < tL = t, we define the operator E (0) (Ω, t, s) = E (0) (tL , tL−1 )E (0) (tL−1 , tL−2 ) · · · E (0) (t1 , t0 ), whose integral kernel e(0) (Ω, t, s)(x, y) can be explicitly computed from (8). The parametrix E (0) (Ω, t, s) is then expected to converge (in some sense) to the actual propagator U (t, s) in the limit ω(Ω) = max{tj − tj −1 , j = 1, . . . , L} → 0.

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183

We have not specified the path along which the action functional in (8) should be evaluated. A standard choice, inspired by the custom in physics after Feynman [25], is the broken line approximation introduced above in (6), namely γ (τ ) = xj +

xj +1 − xj (τ − tj ), tj +1 − tj

τ ∈ [tj , tj +1 ],

j = 0, . . . , L.

A quite complete theory of path integration in this context has been developed by Kumano-go [43]. In fact, the time-slicing approximation shows its full power when straight lines are replaced by classical paths. To be precise, under the assumptions on the potential detailed above, a short-time analysis of the Schrödinger flow reveals that there exists δ > 0 such that for 0 < |t − s| ≤ δ and any x, y ∈ Rd there exists a unique solution γ of the classical equation of motion γ¨ (τ ) = −∇V (τ, γ (τ )) satisfying the boundary conditions γ (s) = y, γ (t) = x. In particular, this can be adapted to the subdivision Ω by making the separation small enough, namely ω(Ω) ≤ δ. A detailed analysis can be found in [29, Chap. 2]. Among the large number of results proved in this context we mention two milestones from forerunner papers by Fujiwara. In [27] he proved convergence of E (0) (Ω, t, s) to U (t, s) in the norm operator topology in B(L2 (Rd ))—the space of bounded operators in L2 . Under the same hypotheses convergence at the level of integral kernels in a very strong topology was proved in [28]. It should be emphasized that the aforementioned results are given for the higher order parametrices E (N ) (t, s), N ∈ N0 , also known as Birkhoff–Maslov parametrices [6, 49] and defined by  i 1 e h¯ S(t,s,x,y) a (N ) (h, ¯ t, s)(x, y)ϕ(y)dy, d/2 d (2π i h(t − s)) ¯ R (9) 3N i 1−j aj (t, s)(x, y) for suitable functions where a (N ) (h, ¯ t, s)(x, y) = j =1 ( h¯ ) aj (t, s) ∈ Cb∞ (R2d ) for t − s ≤ δ, with a0 (t, s) ≡ 1. We remark that E (N ) (t, s) are parametrices in the sense that they satisfy E (N ) (t, s)ϕ(x) =

2 (N ) (N ) (i h∂ ¯ t + h¯ ./2 − V (t, x))E ψ = G (t, s)ψ,

(10)

but a (N ) is 4replaced by the amplitude function where G(N ) (t, s) has the form in (9) 4 (N (N ) g (h, ¯ t, s)(x, y) which satisfies 4g ) (h, ¯ t, s)4m ≤ Cm h¯ N +1 |t − s|N +1 , m ∈ N0 . As before, the case of a long interval [s, t] can be treated by means of composition over a sufficiently fine subdivision Ω = t0 , . . . , tL of the interval [s, t] such that s = t0 < t1 < . . . < tL = t, namely E (N ) (Ω, t, s) = E (N ) (tL , tL−1 )E (N ) (tL−1 , tL−2 ) · · · E (N ) (t1 , t0 ).

(11)

The core results of the L2 theory for the time-slicing approximation read as follows.

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Theorem 1 Let the potential V satisfy Assumption (A) and fix T > 0. For 0 < t − s ≤ T and any subdivision Ω of the interval [s, t] such that ω(Ω) ≤ δ, the following claims hold. 1. There exists a constant C = C(N, T ) > 0 such that 4 4 4 (N ) 4 4E (Ω, t, s) − U (t, s)4 2 2 ≤ C h¯ N ω(Ω)N +1 (t − s), L →L

N ∈ N0 .

(12)

2. We have (cf. (7)) lim

ω(Ω)→0

a (N ) (Ω, h, ¯ t, s) = a(h, ¯ t, s)

in Cb∞ (R2d ).

Precisely, there exists C = C(m, N, T ) > 0 such that 4 4 4 4 (N ) N N +1 (t − s), 4a(h, ¯ t, s) − a (Ω, h, ¯ t, s)4 ≤ C h¯ ω(Ω) m

m, N ∈ N0 .

The proof of these results ultimately relies on fine analysis of OIOs. The underlying overall strategy can be condensed as follows: 1. prove that “time-slicing approximation is an oscillatory integral” (cf. [29]), i.e., that the operators arising from (9) are indeed OIOs under suitable assumptions; 2. derive precise estimates for the operator norm of such OIOs; 3. employ the algebra property of B(L2 ) in order to deal with composition in (11). With reference to the last item, we mention that an aspect to be considered is that composition of OIOs results in an OIO only for short times, due to the occurrence of caustics, and in general one should not expect smoothing effects for long times. For the sake of completeness we also mention that Nicola showed in [55] how parts of the conclusions in Theorem 1 still hold under weaker regularity assumptions for the potential. Assumption (A) is now replaced by the following one. Assumption (A’) The potential V : R × Rd → R belongs to L1loc (R × Rd ) and for almost every t ∈ R and |α| ≤ 2 the derivatives ∂xα V (t, x) exist and are continuous with respect to x. Furthermore d+1 (Rd )), ∂xα V (t, x) ∈ L∞ (R; Hul

|α| = 2,

n (Rd ), n ∈ N, is the Kato–Sobolev space (also known as uniformly local where Hul Sobolev space) of functions f ∈ L1loc (Rd ) satisfying f H n = supB f H n (B) < ul ∞, the supremum being computed on all open balls B ⊂ Rd of radius 1.

Theorem 2 ([55, Thm. 1.1]) Let the potential V satisfy Assumption (A’). For any T > 0 there exists C = C(T ) > 0 such that for any 0 < t − s ≤ T and any subdivision Ω of the interval [s, t] with ω(Ω) ≤ δ and 0 < h¯ ≤ 1,

A Time–Frequency Analysis Perspective on Feynman Path Integrals

4 4 4 (0) 4 4E (Ω, t, s) − U (t, s)4

L2 →L2

185

≤ Cω(Ω)(t − s).

4 Beyond the L2 Theory via Gabor Analysis In view of the results recalled above it seems that the analysis of convergence of time-slicing approximations of path integrals can be suitably conducted at the level of operators on L2 (Rd ), i.e. in the space B(L2 (Rd )) (usually) equipped with the norm operator topology. It is then natural to wonder whether there exists an Lp analogue of Theorem 1 with p = 2. We cannot expect a naive transposition of the claim for several reasons. First of all, notice that the Schrödinger propagator is not even bounded on Lp (Rd ) for p = 2. The parabolic geometry of its characteristic manifold implies that a peculiar loss of derivative, ultimately due to dispersion, occurs [7, 52]: 4 4 4 4 i h. 4e ¯ f 4

Lp

4 4 4 k/2 4 ≤ C 4(1 − h.) f4 ¯

Lp

k = 2d|1/2 − 1/p|,

,

1 < p < ∞.

On the basis of this observation one is lead to consider the following scale of semiclassical Lp -based Sobolev spaces: for 1 < p < ∞ and k ∈ R define p k/2 f Lp < ∞}. L˜ k (Rd ) = {f ∈ S  (Rd ) : f L˜ p = (1 − h.) ¯ k

p p We set Lk (Rd ) = L˜ k (Rd ) in the case where h¯ = 1. This is indeed a suitable setting for the analysis of Schrödinger operators, in particular for Fourier integral operators arising as Schrödinger propagators associated with quadratic Hamiltonians, cf. [16]. p We are also confronted with another issue: the space of bounded operators L˜ k → p L (or vice versa) is clearly not an algebra under composition. This is a major obstacle for a proficient time-slicing approximation, having in mind the construction of the parametrices E (N ) (Ω, t, s) in (11) and the role of this feature in the L2 setting. A possible solution comes from time–frequency analysis, since all these issues become manageable as soon as one transfers the problem to the phase space setting. The first key results in this context are due to Nicola [54] and read as follows—the notation has been introduced in the previous section.

Theorem 3 ([54, Thm. 1.1]) Assume the condition in Assumption (A) and let 1 < p < ∞, k = 2d|1/2 − 1/p|. 1. For any T > 0 there exists a constant C = C(T ) > 0 such that for all f ∈ S (Rd ), |t − s| ≤ T and 0 < h¯ ≤ 1:

U (t, s)f Lp ≤ C f L˜ p ,

1 < p ≤ 2,

U (t, s)f L˜ p ≤ C f Lp ,

2 ≤ p < ∞.

k

−k

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2. For any T > 0 and N ∈ N0 there exists a constant C = C(T ) > 0 such that for 0 < t − s ≤ T and any subdivision Ω of the interval [s, t] with ω(Ω) ≤ δ, f ∈ S (Rd ) and 0 < h¯ ≤ 1: 4  4 4 (N ) 4 4 E (Ω, t, s) − U (t, s) f 4

Lp

≤C h¯ N ω(Ω)N +1 (t−s) f L˜ p , k

1 < p ≤ 2,

4  4 4 (N ) 4 4 E (Ω, t, s) − U (t, s) f 4 ˜ p ≤C h¯ N ω(Ω)N +1 (t−s) f Lp , 2 ≤ p 0 such that T hm,χ h¯ (t,s) ≤ C a m . 4. Let T ∈ F I Oh¯ (χ ), 1 < p < ∞ and k = 2d|1/p − 1/2|. Then T extends to a p bounded operator T : L˜ k (Rd ) → Lp (Rd ) if 1 < p ≤ 2 and T : Lp (Rd ) → p L˜ −k (Rd ) if 2 ≤ p < ∞. In particular, for m > 2d there exists C > 0 such that ¯ ¯

T L˜ p →Lp ≤ C T hm,χ (1

0 such that E (N )  (t, s) → I (identity op.) for t → s in the strong operator topology on E L2 . (N ) (t, s) is a parametrix in the sense that 2. E   1 2 (N ) (t, s) = G(N ) (t, s), i h∂ ¯ t + h¯ Δ − V (t, x) E 2 G(N ) (t, s)f =

1 (2π i(t − s)h) ¯ d/2

 Rd

i

e h¯

S (N) (t,s,x,y)

gN (t, s, x, y)f (y) dy,

where the amplitude gN satisfies

gN (t, s, ·, ·) M ∞,1 (R2d ) ≤ C (t − s)N , for some C = C(T ) > 0. 3. There exists a constant C = C(T ) > 0 such that (N ) (t, s) − U (t, s) L2 →L2 ≤ C h¯ −1 (t − s)N +1 .

E

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These estimates should be compared with those appearing in Sect. 3.2. Similarly, given a subdivision Ω = t0 , . . . , tL of the interval [s, t] such that s = t0 < t1 < . . . < tL = t, we introduce the long-time composition (N ) (tL , tL−1 )E (N ) (tL−1 , tL−2 ) · · · E (N ) (t1 , t0 ), (N ) (Ω, t, s) = E E and the main result in [56] reads as follows. Theorem 8 ([56, Thm. 1]) Let V satisfy Assumption (B) above. For any T > 0 there exists a constant C = C(T ) > 0 such that, for 0 < t − s ≤ T h, ¯ 0 < h¯ ≤ 1, and any sufficiently fine subdivision Ω of the interval [s, t], we have (N ) (Ω, t, s) − U (t, s) L2 →L2 ≤ Cω(Ω)N .

E

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The proof of Theorem 8 is largely inspired by the proof of Theorem 1. In fact, one can isolate a strategy of general interest which can be applied to suitable operators, cf. [56, Thm. 10].

5.1 The Role of h ¯ We already remarked that Birkhoff–Maslov parametrices (9) enjoy several nice properties, one of them being an increasing semiclassical approximation power— the exponent of h¯ in (12) increases with N. This is of course related to the

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construction of the parametrices, relying on piecewise classical paths. This desirable property is lost when one considers rougher parametrices as those in (18), where the balance weights in favour of accelerated rate of convergence with respect to time. (N ) in Theorem 7 reveals that A cursory glance at the estimates for the operators E negative powers of h¯ are involved, making them completely unfit for semiclassical arguments. Nevertheless, one can also notice that all the estimates are uniform in h¯ as soon as time is measured in units of h¯ , which is a particularly interesting feature.

5.2 The Role of M ∞,1 Although being hidden in the details of the proofs, the role of the Sjöstrand class M ∞,1 (Rd ) is crucial for the results presented insofar. There is in particular a special feature of this space playing a major role in the arguments, namely the fact that it is a commutative Banach algebra under pointwise product. In general, precise p,q conditions on p, q, s, and d must hold in order for Ms (Rd ) to be a Banach algebra with respect to pointwise multiplication4 . Proposition 1 ([60, Thm. 3.5 and Cor. 2.2]) Let 1 ≤ p, q ≤ ∞ and s ∈ R. The following facts are equivalent. p,q

1. Ms (Rd ) is a Banach algebra for pointwise multiplication. p,q 2. Ms (Rd ) #→ L∞ (Rd ). 3. Either s = 0 and q = 1 or s > d/q  .

6 Pointwise Convergence of Integral Kernels A concise way to resume the philosophy behind the operator-theoretic approaches to rigorous path integral discussed in Sect. 3 could be the following one: design suitable sequences of approximation operators and prove that they are bounded together with their compositions, where the latter should converge to the exact propagator in a suitable topology on B(L2 (Rd )). There are good reasons for not being completely satisfied with this state of affairs. First of all, looking back at Feynman’s original paper [23] and the textbook [25] one immediately notices

4 To be precise, the result provided here concerns conditions under which the embedding p,q p,q p,q Ms · Ms #→ Ms is continuous; this means that the algebra property eventually holds up

to a constant. It is well known that one may provide an equivalent norm for which the boundedness estimate holds with C = 1 (cf. [61, Thm. 10.2]). This condition will be tacitly assumed whenever concerned with Banach algebras from now on.

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that the entire process of defining path integrals in (2) can be read in terms of a sequence of integral operators (finite-dimensional approximation operators as in (5) or (9)); in particular, Feynman’s insight calls for the pointwise convergence of their integral kernels to the kernel ut of the propagator. This remark strongly motivates a focus shift from the operators to their kernels, which may appear as an unaffordable problem in general: approximation operators should be first explicitly characterized as integral operators, at least in the sense of distributions by some version of Schwartz’s kernel theorem, then one should determine if the kernels are in fact functions and finally hope for convergence. Both the approximation schemes discussed insofar are well suited for this purpose, since oscillatory integrals are explicitly involved. A clue in this direction, already mentioned at the beginning of the previous section, is that the regularity assumptions in Theorem 1 imply shorttime convergence in a finer topology at the level of integral kernels. The solution of this problem in the framework of the sequential approach as presented in Sect. 3.1 was recently obtained by the author and Nicola in the paper [57], where techniques of time–frequency analysis of functions and operators are heavily used. We need a few preparation in order to state the main result. A word of warning about notation. In this section we restore the “harmonic analysis” normalization of the Fourier transform with 2π in the phase factor. This reflects into the definition of Weyl, Wigner, and short-time Fourier transforms, in contrast with the “PDE” normalization adopted insofar. We are sorry if this choice may cause confusion but the aim is to clean up the relevant formulas from annoying normalization constants. For the same reason we set h¯ = 1 from now on.

6.1 Weyl Operators A summary of the fruitful exchange between analysis of pseudodifferential operators and time–frequency analysis is far beyond the purposes of this note. The crucial point of contact is represented by the Wigner distribution  W (f, g)(x, ξ ) =

Rd

 y  y dy, e−2π iy·ξ f x + g x− 2 2

f, g ∈ S (Rd ),

which is a well-known phase space transform deeply connected with the STFT [18, 30]. We define the Weyl transform σ w : S (Rd ) → S  (Rd ) of the symbol σ ∈ S  (R2d ) by duality as follows: σ w f, g = σ, W (g, f ) ,

f, g ∈ S (Rd ).

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As an elementary example of Weyl operator consider the multiplication by V (x), whose symbol is trivially given by σV (x, ξ ) = V (x) = (V ⊗ 1)(x, ξ ),

(x, ξ ) ∈ R2d .

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The composition of Weyl transforms induces a bilinear form on symbols, the so-called twisted/Weyl product: σ w ◦ ρ w = (σ #ρ)w . Explicit formulas for the twisted product of are known (cf. [69]) but we are more interested in the algebra structure induced on symbol spaces. It turns out that the Sjöstrand M ∞,1 (R2d ), as well as the family of modulation spaces Ms∞ (R2d ) with s > 2d, enjoy a peculiar double Banach algebra structure: • a commutative one associated with pointwise multiplication, as a consequence of Proposition 1; • a non-commutative one associated with the Weyl product of symbols [33, 64]. The latter algebra structure has been thoroughly investigated in view of its role in the distinctive sparse behaviour satisfied by pseudodifferential operators with symbols in those spaces—the so-called almost diagonalization property with respect to time–frequency shifts. Having in mind the Gabor matrix defined in (13), it can be proved that σ ∈ Ms∞ (R2d ) if and only if, for some (hence any) g ∈ S (Rd ) \ {0}, M (σ w , g, z, w) ≤ Cw − z −s ,

z, w ∈ R2d .

Similarly, σ ∈ M ∞,1 (R2d ) if and only if there exists H ∈ L1 (R2d ) such that M (σ w , g, z, w) ≤ H (w − z),

z, w ∈ R2d .

The consequences of phase space sparsity have been thoroughly studied in the papers [11–13, 15, 32, 33], mainly in order to extend Sjöstrand’s theory of Wiener subalgebras of Weyl operators [64] to more general pseudodifferential and Fourier integral operators.

6.2 Main Results In order to state the main results in full generality we need to slightly generalize the free Hamiltonian operator H0 in (1). Let a be a real-valued, time-independent, quadratic homogeneous polynomial on R2d , namely a(x, ξ ) =

1 1 xAx + ξ Bx + ξ Cξ, 2 2

for some symmetric matrices A, C ∈ Rd×d and B ∈ Rd×d . Consider then the Weyl quantization of a as above. A classical result in phase space harmonic analysis (see [18, Sec. 15.1.3] and also [14, 26]) is that the solution of (1) with H0 = a w and V = 0 is given by ψ(t, x) = e−itH0 ϕ(x) = μ(At )ϕ(x),

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where μ(At ) is a metaplectic operator, designed as follows. First, the classical phase space flow governed by the Hamilton equations5  2π z˙ = J ∇z a(z) = A,

A=

B C −A −B 0

 ∈ sp(d, R),

defines a mapping  R t → At = e

(t/2π )A

=

At Bt Ct D t

 ∈ Sp(d, R).

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In very sloppy terms, the metaplectic map μ is a double-valued unitary representation of the symplectic group on L2 (Rd ), hence the classical flow At is “lifted” to a family of unitary operators on L2 (Rd ). Under certain circumstances an explicit characterization for μ(At ) can be provided: for all t ∈ R such that At is a free symplectic matrix, namely such that the upper-right block Bt is invertible, the corresponding metaplectic operator is a quadratic Fourier transform—cf. [18, Sec. 7.2.2]:  μ(At )φ(x) = ct |det Bt |−1/2 e2π iΦt (x,y) φ(y)dy, φ ∈ S (Rd ), (24) Rd

for some ct ∈ C, |ct | = 1, where Φt (x, y) =

1 1 xDt Bt−1 x − yBt−1 x + yBt−1 At y, 2 2

x, y ∈ Rd .

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It is known that H0 is a self-adjoint operator on its domain (see [35]) D (H0 ) = {ψ ∈ L2 (Rd ) : H0 ψ ∈ L2 (Rd )}. In order for the machinery developed in Sect. 3.1 to hold in the case where H0 = a w as above we need to consider a version of Trotter formula which holds for semigroups in more general frameworks (cf., for instance, [19, Cor. 2.7]). For our purposes, it is enough to assume that V is a bounded perturbation of H0 , namely V ∈ B(L2 (Rd )); notice that V ∈ L∞ (Rd ) is then a suitable choice, hence including complex-valued potentials. Therefore, under the hypotheses on H0 and V discussed insofar, we have e−it(H0 +V ) = lim En (t), n→∞

5 The

n  t t En (t) = e−i n H0 e−i n V ,

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factor 2π derives from the normalization of the Fourier transform adopted in this section.

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with convergence in the strong operator topology in L2 (Rd ). Let us denote by en,t (x, y) the distribution kernel of En (t) and by ut (x, y) that of U (t) = e−it(H0 +V ) . We assumed V ∈ L∞ (Rd ), hence there is some room left for tuning the regularity. A suitable choice is given by the modulation spaces Ms∞ (Rd ), s > d, and M ∞,1 (Rd ), also in view of the rich algebraic structure already discussed. In order to grasp the regularity of functions in this space, recall the definition of the Fourier–Lebesgue space: for s ∈ R we set  f ∈ F L1s (Rd )

⇔

f F L1s =

Rd

|F f (ξ )| ξ s dξ < ∞.

The analytic properties of the involved potentials are briefly collected in the following result. Proposition 2 1. 2. 3. 4. 5.

∩s>0 Ms∞ (Rd ) = Cb∞ (Rd ). Ms∞ (Rd ) ⊂ M ∞,1 (Rd ) for s > d. M ∞,1 (Rd ) ⊂ (F L1 )loc (Rd ) ∩ L∞ (Rd ) ⊂ C 0 (Rd ) ∩ L∞ (Rd ). (M ∞,1 )loc (Rd ) = (F L1 )loc (Rd ). F M (Rd ) ⊂ M ∞,1 (Rd ), where F M (Rd ) is the space of Fourier transforms of (finite) complex measures on Rd . Roughly speaking, we have a scale of decreasing regularity spaces.

1. The first is “the best of all possible worlds”, that is Cb∞ (Rd ). 2. At an intermediate stage we have the scale of modulation spaces Ms∞ (Rd ), s > d, populated by bounded continuous functions with decreasing (fractional) regularity as s 1 d. 3. Finally, we have the maximal space M ∞,1 (Rd ) where fractional differentiability is completely lost. Nevertheless it is a space of bounded continuous functions locally enjoying the mild regularity of a function in F L1 . We first present our main result for potentials in Ms∞ (Rd ). Theorem 9 Let H0 = a w as discussed above and V ∈ Ms∞ (Rd ), with s > 2d. Let At denote the classical flow associated with H0 as in (23). For any t ∈ R such that At is free, that is det Bt = 0: 1. the distributions e−2π iΦt en,t , n ≥ 1, and e−2π iΦt ut belong to a bounded subset of Ms∞ (R2d );   2. en,t → ut in F L1r loc (R2d ) for any 0 < r < s − 2d, hence uniformly on compact subsets. The first part of the claim assures that the kernel convergence problem is well posed in this case—the “amplitudes” are bounded continuous functions. The second part precisely characterizes the regularity at which convergence occurs, hence the desired pointwise convergence.

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In view of%the first item  in Proposition 2 and the related characterization C ∞ (R2d ) = r>0 F L1r loc (R2d ), we expect to improve the convergence result in the smooth scenario. Corollary 1 Let H0 = a w as discussed above and V ∈ Cb∞ (Rd ). Let At denote the classical flow associated with H0 as in (23). For any t ∈ R such that At is free, that is det Bt = 0: 1. the distributions e−2π iΦt en,t , n ≥ 1, and e−2π iΦt ut belong to a bounded subset of Cb∞ (R2d ); 2. en,t → ut in C ∞ (R2d ), hence uniformly on compact subsets together with any derivatives. This result should be compared with the second claim in Theorem 1 by Fujiwara, which motivated our quest. In spite of the different assumptions and approximation schemes, we stress that our result is almost global in time–more on exceptional times below. We conclude with the analogous convergence result for potentials in the Sjöstrand class. Theorem 10 Let H0 = a w as discussed above and V ∈ M ∞,1 (Rd ). Let At denote the classical flow associated with H0 as in (23). For any t ∈ R such that At is free, that is det Bt = 0: 1. the distributions e−2π iΦt en,t , n ≥ 1, and e−2π iΦt ut belong to a bounded subset of M ∞,1 (R2d);  2. en,t → ut in F L1 loc (R2d ), hence uniformly on compact subsets. It seems appropriate to highlight that a typical potential setting in the papers by Albeverio and coauthors [1–3, 5] and Itô [38, 39] is “harmonic oscillator plus a bounded perturbation”, the latter in the form of the Fourier transform of a finite complex measure on Rd . While the cited references rely on completely different mathematical schemes for path integrals (which are manufactured as infinitedimensional OIOs), in view of the embedding F M (Rd ) ⊂ M ∞,1 (Rd ) mentioned in Proposition 2 we are able to encompass this class of potentials too.

6.3 The Proof at a Glance In order to understand why our choice of modulation spaces is suitable for the purpose of pointwise convergence we outline the general strategy of the proof of Theorem 9. The first step is to express the parametrix En (t) in integral form and derive a manageable form of the kernel en,t . The algebra property of Ms∞ (Rd ) will play a crucial role from now on. First, we are able to write

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n  n     t −i nt H0 −i nt V En (t) = e e = μ At/n 1 + i V0 n for a suitable V0 = V0,n,t ∈ Ms∞ (Rd )–see [57, Lem. 3.2]. We now expand the (ordered) product and identify multiplication by 1 + itV0 /n with a suitable Weyl operator, then the nice intertwining properties of Weyl and metaplectic operators (the so-called symplectic covariance of Weyl calculus, cf. [18, Thm. 215]) yield ) En (t) =

n  >

k=1

w t  I +i σV0 ◦ A−k t n n

+

n  w μ(At ), μ At/n = an,t

where the first (ordered) product is understood in 4 the 4 Banach algebra w satisfies the estimate 4a 4 C(t)t for (Ms∞ (Rd , ), #). The symbol of an,t n,t M ∞ ≤ e s some locally bounded constant C(t) > 0 independent of n. Since At is a free symplectic matrix, the integral formula (24) holds and with the help of some technical lemmas we are able to precisely characterize the integral kernels en,t and ut as temperate distributions. The non-trivial step is to prove convergence in S  (R2d ), but it can be handled with Banach algebras techniques and some topological arguments. The assumptions on potentials imply the boundedness of {en,t } in Ms∞ (R2d ). Finally, the convergence of en,t to ut in  the1  sequence F Lr loc (R2d ), 0 < r < s − 2d, essentially follow by dominated convergence arguments. The proof of Theorem 10 ultimately moves along the same lines but is more involved, so we will not give the details here. The basic ingredient is a “high-cut filter decomposition” of M ∞,1 , see [57, Lem. 3.3]: a rough function f ∈ M ∞,1 (Rd ) can be splitted as a sum of a very regular part f1 ∈ Cb∞ (Rd ) plus an arbitrarily small (in norm) rough remainder f2 ∈ M ∞,1 (Rd ). Theorem 10 essentially follows as a perturbation of Theorem 9 after a careful management of these remainders.

6.4 Why Exceptional Times? The occurrence of a set of exceptional times in Theorems 9 and 10 is to be expected from a mathematical point of view: it may happen that the integral kernel of the propagator degenerates into a distribution. A well-known example of this behaviour is provided by the harmonic oscillator, already met in Sect. 4.3. Mehler’s formula (16) precisely shows the expected degenerate behaviour, which is consistent with the fact that At is free if and only if t = kπ , k ∈ Z, since Bt = (sin t)Id×d (up to normalization constants). The physical interpretation of the exceptional values is not entirely clear at the moment, but milder convergence results in the spirit of the theorems above may be proved to hold also at exceptional times. These and other related issues will be object of forthcoming contributions [20].

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Applied and Numerical Harmonic Analysis (101 volumes)

1. A. I. Saichev and W. A. Woyczy´nski: Distributions in the Physical and Engineering Sciences (ISBN: 978-0-8176-3924-2) 2. C. E. D’Attellis and E. M. Fernandez-Berdaguer: Wavelet Theory and Harmonic Analysis in Applied Sciences (ISBN: 978-0-8176-3953-2) 3. H. G. Feichtinger and T. Strohmer: Gabor Analysis and Algorithms (ISBN: 978-0-8176-3959-4) 4. R. Tolimieri and M. An: Time-Frequency Representations (ISBN: 978-0-81763918-1) 5. T. M. Peters and J. C. Williams: The Fourier Transform in Biomedical Engineering (ISBN: 978-0-8176-3941-9) 6. G. T. Herman: Geometry of Digital Spaces (ISBN: 978-0-8176-3897-9) 7. A. Teolis: Computational Signal Processing with Wavelets (ISBN: 978-08176-3909-9) 8. J. Ramanathan: Methods of Applied Fourier Analysis (ISBN: 978-0-81763963-1) 9. J. M. Cooper: Introduction to Partial Differential Equations with MATLAB (ISBN: 978-0-8176-3967-9) 10. Procházka, N. G. Kingsbury, P. J. Payner, and J. Uhlir: Signal Analysis and Prediction (ISBN: 978-0-8176-4042-2) 11. W. Bray and C. Stanojevic: Analysis of Divergence (ISBN: 978-1-4612-74674) 12. G. T. Herman and A. Kuba: Discrete Tomography (ISBN: 978-0-8176-4101-6) 13. K. Gröchenig: Foundations of Time-Frequency Analysis (ISBN: 978-0-81764022-4) 14. L. Debnath: Wavelet Transforms and Time-Frequency Signal Analysis (ISBN: 978-0-8176-4104-7) 15. J. J. Benedetto and P. J. S. G. Ferreira: Modern Sampling Theory (ISBN: 978-0-8176-4023-1)

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16. D. F. Walnut: An Introduction to Wavelet Analysis (ISBN: 978-0-8176-39624) 17. A. Abbate, C. DeCusatis, and P. K. Das: Wavelets and Subbands (ISBN: 978-0-8176-4136-8) 18. O. Bratteli, P. Jorgensen, and B. Treadway: Wavelets Through a Looking Glass (ISBN: 978-0-8176-4280-80 19. H. G. Feichtinger and T. Strohmer: Advances in Gabor Analysis (ISBN: 978-0-8176-4239-6) 20. O. Christensen: An Introduction to Frames and Riesz Bases (ISBN: 978-08176-4295-2) 21. L. Debnath: Wavelets and Signal Processing (ISBN: 978-0-8176-4235-8) 22. G. Bi and Y. Zeng: Transforms and Fast Algorithms for Signal Analysis and Representations (ISBN: 978-0-8176-4279-2) 23. J. H. Davis: Methods of Applied Mathematics with a MATLAB Overview (ISBN: 978-0-8176-4331-7) 24. J. J. Benedetto and A. I. Zayed: Sampling, Wavelets, and Tomography (ISBN: 978-0-8176-4304-1) 25. E. Prestini: The Evolution of Applied Harmonic Analysis (ISBN: 978-0-81764125-2) 26. L. Brandolini, L. Colzani, A. Iosevich, and G. Travaglini: Fourier Analysis and Convexity (ISBN: 978-0-8176-3263-2) 27. W. Freeden and V. Michel: Multiscale Potential Theory (ISBN: 978-0-81764105-4) 28. O. Christensen and K. L. Christensen: Approximation Theory (ISBN: 978-08176-3600-5) 29. O. Calin and D.-C. Chang: Geometric Mechanics on Riemannian Manifolds (ISBN: 978-0-8176-4354-6) 30. J. A. Hogan: Time?Frequency and Time?Scale Methods (ISBN: 978-0-81764276-1) 31. C. Heil: Harmonic Analysis and Applications (ISBN: 978-0-8176-3778-1) 32. K. Borre, D. M. Akos, N. Bertelsen, P. Rinder, and S. H. Jensen: A SoftwareDefined GPS and Galileo Receiver (ISBN: 978-0-8176-4390-4) 33. T. Qian, M. I. Vai, and Y. Xu: Wavelet Analysis and Applications (ISBN: 978-3-7643-7777-9) 34. G. T. Herman and A. Kuba: Advances in Discrete Tomography and Its Applications (ISBN: 978-0-8176-3614-2) 35. M. C. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott: Advances in Mathematical Finance (ISBN: 978-0-8176-4544-1) 36. O. Christensen: Frames and Bases (ISBN: 978-0-8176-4677-6) 37. P. E. T. Jorgensen, J. D. Merrill, and J. A. Packer: Representations, Wavelets, and Frames (ISBN: 978-0-8176-4682-0) 38. M. An, A. K. Brodzik, and R. Tolimieri: Ideal Sequence Design in TimeFrequency Space (ISBN: 978-0-8176-4737-7) 39. S. G. Krantz: Explorations in Harmonic Analysis (ISBN: 978-0-8176-4668-4)

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40. B. Luong: Fourier Analysis on Finite Abelian Groups (ISBN: 978-0-81764915-9) 41. G. S. Chirikjian: Stochastic Models, Information Theory, and Lie Groups, Volume 1 (ISBN: 978-0-8176-4802-2) 42. C. Cabrelli and J. L. Torrea: Recent Developments in Real and Harmonic Analysis (ISBN: 978-0-8176-4531-1) 43. M. V. Wickerhauser: Mathematics for Multimedia (ISBN: 978-0-8176-48794) 44. B. Forster, P. Massopust, O. Christensen, K. Gröchenig, D. Labate, P. Vandergheynst, G. Weiss, and Y. Wiaux: Four Short Courses on Harmonic Analysis (ISBN: 978-0-8176-4890-9) 45. O. Christensen: Functions, Spaces, and Expansions (ISBN: 978-0-8176-49791) 46. J. Barral and S. Seuret: Recent Developments in Fractals and Related Fields (ISBN: 978-0-8176-4887-9) 47. O. Calin, D.-C. Chang, and K. Furutani, and C. Iwasaki: Heat Kernels for Elliptic and Sub-elliptic Operators (ISBN: 978-0-8176-4994-4) 48. C. Heil: A Basis Theory Primer (ISBN: 978-0-8176-4686-8) 49. J. R. Klauder: A Modern Approach to Functional Integration (ISBN: 978-08176-4790-2) 50. J. Cohen and A. I. Zayed: Wavelets and Multiscale Analysis (ISBN: 978-08176-8094-7) 51. D. Joyner and J.-L. Kim: Selected Unsolved Problems in Coding Theory (ISBN: 978-0-8176-8255-2) 52. G. S. Chirikjian: Stochastic Models, Information Theory, and Lie Groups, Volume 2 (ISBN: 978-0-8176-4943-2) 53. J. A. Hogan and J. D. Lakey: Duration and Bandwidth Limiting (ISBN: 9780-8176-8306-1) 54. G. Kutyniok and D. Labate: Shearlets (ISBN: 978-0-8176-8315-3) 55. P. G. Casazza and P. Kutyniok: Finite Frames (ISBN: 978-0-8176-8372-6) 56. V. Michel: Lectures on Constructive Approximation (ISBN: 978-0-81768402-0) 57. D. Mitrea, I. Mitrea, M. Mitrea, and S. Monniaux: Groupoid Metrization Theory (ISBN: 978-0-8176-8396-2) 58. T. D. Andrews, R. Balan, J. J. Benedetto, W. Czaja, and K. A. Okoudjou: Excursions in Harmonic Analysis, Volume 1 (ISBN: 978-0-8176-8375-7) 59. T. D. Andrews, R. Balan, J. J. Benedetto, W. Czaja, and K. A. Okoudjou: Excursions in Harmonic Analysis, Volume 2 (ISBN: 978-0-8176-8378-8) 60. D. V. Cruz-Uribe and A. Fiorenza: Variable Lebesgue Spaces (ISBN: 978-30348-0547-6) 61. W. Freeden and M. Gutting: Special Functions of Mathematical (Geo-)Physics (ISBN: 978-3-0348-0562-9) 62. A. I. Saichev and W. A. Woyczy´nski: Distributions in the Physical and Engineering Sciences, Volume 2: Linear and Nonlinear Dynamics of Continuous Media (ISBN: 978-0-8176-3942-6)

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63. S. Foucart and H. Rauhut: A Mathematical Introduction to Compressive Sensing (ISBN: 978-0-8176-4947-0) 64. G. T. Herman and J. Frank: Computational Methods for Three-Dimensional Microscopy Reconstruction (ISBN: 978-1-4614-9520-8) 65. A. Paprotny and M. Thess: Realtime Data Mining: Self-Learning Techniques for Recommendation Engines (ISBN: 978-3-319-01320-6) 66. A. I. Zayed and G. Schmeisser: New Perspectives on Approximation and Sampling Theory: Festschrift in Honor of Paul Butzer’s 85th Birthday (ISBN: 978-3-319-08800-6) 67. R. Balan, M. Begue, J. Benedetto, W. Czaja, and K. A. Okoudjou: Excursions in Harmonic Analysis, Volume 3 (ISBN: 978-3-319-13229-7) 68. H. Boche, R. Calderbank, G. Kutyniok, and J. Vybiral: Compressed Sensing and its Applications (ISBN: 978-3-319-16041-2) 69. S. Dahlke, F. De Mari, P. Grohs, and D. Labate: Harmonic and Applied Analysis: From Groups to Signals (ISBN: 978-3-319-18862-1) 70. A. Aldroubi: New Trends in Applied Harmonic Analysis (ISBN: 978-3-31927871-1) 71. M. Ruzhansky: Methods of Fourier Analysis and Approximation Theory (ISBN: 978-3-319-27465-2) 72. G. Pfander: Sampling Theory, a Renaissance (ISBN: 978-3-319-19748-7) 73. R. Balan, M. Begue, J. Benedetto, W. Czaja, and K. A. Okoudjou: Excursions in Harmonic Analysis, Volume 4 (ISBN: 978-3-319-20187-0) 74. O. Christensen: An Introduction to Frames and Riesz Bases, Second Edition (ISBN: 978-3-319-25611-5) 75. E. Prestini: The Evolution of Applied Harmonic Analysis: Models of the Real World, Second Edition (ISBN: 978-1-4899-7987-2) 76. J. H. Davis: Methods of Applied Mathematics with a Software Overview, Second Edition (ISBN: 978-3-319-43369-1) 77. M. Gilman, E. M. Smith, and S. M. Tsynkov: Transionospheric Synthetic Aperture Imaging (ISBN: 978-3-319-52125-1) 78. S. Chanillo, B. Franchi, G. Lu, C. Perez, and E. T. Sawyer: Harmonic Analysis, Partial Differential Equations and Applications (ISBN: 978-3-319-52741-3) 79. R. Balan, J. Benedetto, W. Czaja, M. Dellatorre, and K. A. Okoudjou: Excursions in Harmonic Analysis, Volume 5 (ISBN: 978-3-319-54710-7) 80. I. Pesenson, Q. T. Le Gia, A. Mayeli, H. Mhaskar, and D. X. Zhou: Frames and Other Bases in Abstract and Function Spaces: Novel Methods in Harmonic Analysis, Volume 1 (ISBN: 978-3-319-55549-2) 81. I. Pesenson, Q. T. Le Gia, A. Mayeli, H. Mhaskar, and D. X. Zhou: Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science: Novel Methods in Harmonic Analysis, Volume 2 (ISBN: 978-3-319-55555-3) 82. F. Weisz: Convergence and Summability of Fourier Transforms and Hardy Spaces (ISBN: 978-3-319-56813-3)

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83. C. Heil: Metrics, Norms, Inner Products, and Operator Theory (ISBN: 978-3319-65321-1) 84. S. Waldron: An Introduction to Finite Tight Frames: Theory and Applications. (ISBN: 978-0-8176-4814-5) 85. D. Joyner and C. G. Melles: Adventures in Graph Theory: A Bridge to Advanced Mathematics. (ISBN: 978-3-319-68381-2) 86. B. Han: Framelets and Wavelets: Algorithms, Analysis, and Applications (ISBN: 978-3-319-68529-8) 87. H. Boche, G. Caire, R. Calderbank, M. März, G. Kutyniok, and R. Mathar: Compressed Sensing and Its Applications (ISBN: 978-3-319-69801-4) 88. A. I. Saichev and W. A. Woyczy´nski: Distributions in the Physical and Engineering Sciences, Volume 3: Random and Fractal Signals and Fields (ISBN: 978-3-319-92584-4) 89. G. Plonka, D. Potts, G. Steidl, and M. Tasche: Numerical Fourier Analysis (978-3-030-04305-6) 90. K. Bredies and D. Lorenz: Mathematical Image Processing (ISBN: 978-3030-01457-5) 91. H. G. Feichtinger, P. Boggiatto, E. Cordero, M. de Gosson, F. Nicola, A. Oliaro, and A. Tabacco: Landscapes of Time-Frequency Analysis (ISBN: 9783-030-05209-6) 92. E. Liflyand: Functions of Bounded Variation and Their Fourier Transforms (978-3-030-04428-2) 93. R. Campos: The XFT Quadrature in Discrete Fourier Analysis (978-3-03013422-8) 94. M. Abell, E. Iacob, A. Stokolos, S. Taylor, S. Tikhonov, J. Zhu: Topics in Classical and Modern Analysis: In Memory of Yingkang Hu (978-3-03012276-8) 95. H. Boche, G. Caire, R. Calderbank, G. Kutyniok, R. Mathar, P. Petersen: Compressed Sensing and its Applications: Third International MATHEON Conference 2017 (978-3-319-73073-8) 96. A. Aldroubi, C. Cabrelli, S. Jaffard, U. Molter: New Trends in Applied Harmonic Analysis, Volume II: Harmonic Analysis, Geometric Measure Theory, and Applications (978-3-030-32352-3) 97. S. Dos Santos, M. Maslouhi, K. Okoudjou: Recent Advances in Mathematics and Technology: Proceedings of the First International Conference on Technology, Engineering, and Mathematics, Kenitra, Morocco, March 26-27, 2018 (978-3-030-35201-1) 98. Á. Bényi, K. Okoudjou: Modulation Spaces: With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations (978-1-0716-03307) 99. P. Boggiato, M. Cappiello, E. Cordero, S. Coriasco, G. Garello, A. Oliaro, J. Seiler: Advances in Microlocal and Time-Frequency Analysis (978-3-03036137-2)

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100. S. Casey, K. Okoudjou, M. Robinson, B. Sadler: Sampling: Theory and Applications (978-3-030-36290-4) 101. P. Boggiatto, T. Bruno, E. Cordero, H. G. Feichtinger, F. Nicola, A. Oliaro, A. Tabacco, M. Vallarino: Landscapes of Time-Frequency Analysis (978-3030-56004-1) For an up-to-date list of ANHA titles, please visit http://www.springer.com/ series/4968