Convolution Transform 9781400877072

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THE CONVOLUTION TRANSFORM

PRINCETON MATHEMATICAL SERIES

Editors: MABSTON MOBSE and A. W. TUCKEB 1. The Classical Groups, Their Invariants and Representations.

By

HERMANN WEYL.

2. Topological Groups. By L. PONTRJAGIN. Translated by EMMA LEHMEB. 3. An Introduction to Differential Geometry with Use of the Tensor Calculus. By LTTTHEB PFAHLEB EISENHABT. 4. Dimension Theory. By WITOLD HUBEWICZ and HENBY WALLMAN. 5. The Analytical Foundations of Celestial Mechanics. By AUBEL WINTNEB. 6. The Laplace Transform. By DAVID VEBNON WIDDEB. 7. Integration. By EDWABD JAMES MCSHANE. 8. Theory of Lie Groups: I. By CLAUDE CHEVALLEY. 9. Mathematical Methods of Statistics. By HABALD CRAMISB. 10. Several Complex Variables. By S. BOCHNEB and W. T. MABTIN. 11. Introduction to Topology. By SOLOMON LEFSCHETZ. 12. The Topology of Surfaces and their Transformations. By JAKOB NIELSEN and WEBNEB FENCHEL. 13. Algebraic Curves. By ROBEBT J. WALKEB. 14. The Topology of Fibre Bundles. By NOBMAN STEENBOD. 15. Foundations of Algebraic Topology. By SAMUEL EILENBEBO and NOBMAN STEENBOD.

16. Functionals of Finite Riemann Surfaces. By MENAHEM SCHIFFEB and DONALD C. SPENCER.

17. Introduction to Mathematical Logic, Vol. I. By ALONZO CHUBCH. 18. Algebraic Geometry. By S. LEFSCHETZ. 19. Homological Algebra. By HENBI CABTAN and SAMUEL EILENBEBG. 20. The Convolution Transform. By I. I. HIBSCHMAN and D. V. WIDDEB

THE CONVOLUTION TRANSFORM BY

I. I. HIRSCHMAN AND

D. V. WIDDER

1955

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Published, 1955, by Princeton University Press London: Geoffrey Cumberlege, Oxford University Press L.C. CARD 54—6080

COMPOSITION BY THE PITMAN PEESS, BELFAST PRINTED IN THE UNITED STATES Or AMEKICA

Preface THE operation of convolution applied to sequences or functions is basic in analysis. It arises when two power series or two Laplace (or Fourier) integrals are multiplied together. Also most of the classical integral transforms involve integrals which define convolutions. For the present authors the convolution transform came as a natural generalization of the Laplace transform. It was early recognized that the now familiar real inversion of the latter is essentially accomplished by a particular linear differential operator of infinite order (in which translations are allowed). When one studies general operators of the same nature one encounters immediately general convolution transforms as the objects which they invert. This relation between differential operators and integral trans­ forms is the basic theme of the present study. The book may be read easily by anyone who has a working knowledge of real and complex variable theory. For such a reader it should be complete in itself, except that certain fundamentals from The Laplace Transform (number 6 in this series) are assumed. However, it is by no means necessary to have read that treatise completely in order to under­ stand this one. Indeed some of those earlier results can now be better understood as special cases of the newer developments. In conclusion we wish to thank the editors of the Princeton Mathe­ matical Series for including this book in the series. I. I. HIRSCHMAN D. V. WIDDER

Contents CHAPTER I INTRODUCTION SECTION

1. 2. 3. 4. 5. 6. 7. 8. 9.

PAGE

Introduction Convolutions Operational calculus Green's functions Operational calculus continued Thegenerationofkernels Variation diminishing convolutions Outline of program Summary

3 3 5 7 8 11 12 14 16

CHAPTER II THE FINITE KERNELS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Introduction Distribution functions Frequency functions Characteristic functions Convolutions The finite kernels Inversion Exponential polynomials Green's functions Examples Summary

17 17 19 20 22 24 28 30 32 35 36

CHAPTER III THE NON-FINITE KERNELS 1. 2. 3. 4. 5.

Introduction Limits of distribution functions Polya's class of entire functions The closure of a class of distribution functions The non-finite kernels vii

37 38 42 48 49

CONTENTS

Vtii SECTION

6. 7. 8. 9. 10. 11.

PAGE

Properties of the non-finite kernels Inversion Green's functions Examples Associated kernels Summary

55 56 59 65 79 82

CHAPTEB IV

VARIATION DIMINISHING TRANSFORMS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Introduction Generation of variation diminishing frequency functions Logarithmic convexity Characterization of variation diminishing functions The changes of sign of G„} are real and

as a possible inversion formula for (5).

and this suggests

10

INTRODUCTION

D.

[CH. I

If

(7)

then the formula (4) implies that

and since, see Titohmarsh [1939; 257], we (8) may conjecture that Other expansions for cos m and would lead to different "definitions" of cos itD and The product definitions given here are characteristic of our theory. If suitable choices are made for bn then the formulas (6) and (8) become, after a change of variables, well known operational inversion formulas for the Stieltjes and Laplace transforms. Let us verify this in detail for the Laplace transform. We have shown in § 2 that if in

we put

then we obtain

If we define

§ 6]

THE GENERATION

OF KERNELS

U

Making use of (9) we see that (10) is equivalent to the familiar inversion formula

see D. V. Widder [1946; 288]. For a similar discussion of the Stieltjes transform see § 9 of Chapter III. 6. THE GENERATION OF KERNELS

6.1.

be real numbers such that

(1) We define (2)

The condition (1) insures that the infinite product (2) is convergent. See E. C. Titchmarsh [1939; 250]. If there exists a function 0(t) such that (3) then the considerations of the preceding section suggest strongly that the convolution transform (4)

10

[CH. I

INTRODUCTION

is inverted by the formula (5) where

Here the bn are real numbers such that The complex inversion formula for the bilateral Laplace transform asserts that if the transform

converges absolutely in the strip restrictions)

then (under certain

We therefore set (6) We shall ultimately prove that 0(t) defined by (6) satisfies (3) and that the convolution transform (4) is indeed inverted by the operational inversion formula (5). In point of fact we shall treat a slightly more general class of kernels. Le be real and such that (7) and let

It is to the study of the kernels (8) and their associated convolution transforms that the present book is devoted. 7. VARIATION DIMINISHING CONVOLUTIONS

7.1. It is natural to ask why when our operational procedures apply, at least formally, to every convolution transform we have limited ourselves to the kernels 6.1(8). The reasons for this lie somewhat deeper than the operational calculus, and depend upon the following result.

§ 7]

VARIATION

DIMINISHING

CONVOLUTIONS

13

where