Exploring Mathematical Analysis, Approximation Theory, and Optimization: 270 Years Since A.-M. Legendre’s Birth (Springer Optimization and Its Applications, 207) 3031464869, 9783031464867

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Springer Optimization and Its Applications  207

Nicholas J. Daras Michael Th. Rassias Nikolaos B. Zographopoulos   Editors

Exploring Mathematical Analysis, Approximation Theory, and Optimization 270 Years Since A.-M. Legendre’s Birth

Springer Optimization and Its Applications Volume 207 Series Editors Panos M. Pardalos , University of Florida My T. Thai , University of Florida Honorary Editor Ding-Zhu Du, University of Texas at Dallas Advisory Editors Roman V. Belavkin, Middlesex University John R. Birge, University of Chicago Sergiy Butenko, Texas A&M University Vipin Kumar, University of Minnesota Anna Nagurney, University of Massachusetts Amherst Jun Pei, Hefei University of Technology Oleg Prokopyev, University of Pittsburgh Steffen Rebennack, Karlsruhe Institute of Technology Mauricio Resende, Amazon Tamás Terlaky, Lehigh University Van Vu, Yale University Michael N. Vrahatis, University of Patras Guoliang Xue, Arizona State University Yinyu Ye, Stanford University

Aims and Scope Optimization has continued to expand in all directions at an astonishing rate. New algorithmic and theoretical techniques are continually developing and the diffusion into other disciplines is proceeding at a rapid pace, with a spot light on machine learning, artificial intelligence, and quantum computing. Our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in areas not limited to applied mathematics, engineering, medicine, economics, computer science, operations research, and other sciences. The series Springer Optimization and Its Applications (SOIA) aims to publish state-of-the-art expository works (monographs, contributed volumes, textbooks, handbooks) that focus on theory, methods, and applications of optimization. Topics covered include, but are not limited to, nonlinear optimization, combinatorial optimization, continuous optimization, stochastic optimization, Bayesian optimization, optimal control, discrete optimization, multi-objective optimization, and more. New to the series portfolio include Works at the intersection of optimization and machine learning, artificial intelligence, and quantum computing. Volumes from this series are indexed by Web of Science, zbMATH, Mathematical Reviews, and SCOPUS.

Nicholas J. Daras • Michael Th. Rassias • Nikolaos B. Zographopoulos Editors

Exploring Mathematical Analysis, Approximation Theory, and Optimization 270 Years Since A.-M. Legendre’s Birth

Editors Nicholas J. Daras Department of Mathematics and Engineering Sciences Hellenic Military Academy Vari Attikis, Greece Nikolaos B. Zographopoulos Department of Mathematics and Engineering Sciences Hellenic Military Academy Vari Attikis, Greece

Michael Th. Rassias Department of Mathematics and Engineering Sciences Hellenic Military Academy Vari Attikis, Greece Institute for Advanced Study Program in Interdisciplinary Studies Princeton NJ, USA

ISSN 1931-6828 ISSN 1931-6836 (electronic) Springer Optimization and Its Applications ISBN 978-3-031-46486-7 ISBN 978-3-031-46487-4 (eBook) https://doi.org/10.1007/978-3-031-46487-4 Mathematics Subject Classification: 26-XX, 30-XX, 33-XX, 34-XX, 35-XX, 37-XX, 39-XX, 40-XX, 41-XX, 42-XX, 43-XX, 44-XX, 46-XX, 47-XX, 49-XX, 52-XX, 53-XX, 55-XX, 68-XX, 90-XX © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The volume Exploring Mathematical Analysis, Approximation Theory, and Optimization publishes research and survey papers devoted to a broad spectrum of areas of Mathematical Analysis, Approximation Theory and Optimization. As this volume is dedicated to the 270 years since A.-M. Legendre’s birth, effort has been made for the contributed chapters to be devoted to branches that have been influenced, directly or indirectly, by A.-M. Legendre’s tremendous contributions to Mathematics and its applications. Additionally, articles providing a historical background related to Legendre’s work and its association to the foundation of Greece’s higher education are also presented. Topics treated within this volume include the investigation of the JensenSteffensen inequality, Ostrowski and trapezoid type inequalities, a Hilbert-Type Inequality, Hardy’s inequality, dynamic unilateral contact problems, square-free values of a category of integers, a maximum principle for general nonlinear operators, the application of Ergodic Theory to an alternating series expansion for real numbers, bounds for similarity condition numbers of unbounded operators, finite element methods with higher order polynomials, generating functions for the Fubini type polynomials, local asymptotics for orthonormal polynomials, trends in geometric function theory, quasi-variational inclusions, Kleene fixed point theorems, ergodic states, spontaneous symmetry breaking and quasi-averages. We express our gratitude to all the contributing authors of this volume, who have participated in this collective effort. We would also like to express our warmest thanks to the Springer staff for their valuable help throughout the publication process of this work. Vari Attikis, Greece

Nicholas J. Daras Michael Th. Rassias Nikolaos B. Zographopoulos

v

Contents

On a Version of Jensen-Steffensen Inequality and a Note on Inequalities in Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shoshana Abramovich

1

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oanh Chau, Adrien Petrov and Arnaud Heibig

17

Square-Free Values of .[nc tanθ (log n)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. I. Dimitrov

33

Ostrowski and Trapezoid Type Inequalities for Riemann-Liouville Fractional Integrals of Functions with Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silvestru Sever Dragomir A Strong Maximum Principle for General Nonlinear Operators. . . . . . . . . . . Lucas Fresse and Viorica V. Motreanu On the Application of Ergodic Theory to an Alternating Series Expansion for Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chryssoula Ganatsiou and Ilias K. Savvas

45 67

85

Bounds for Similarity Condition Numbers of Unbounded Operators . . . . . 101 Michael Gil’ Legendre’s Geometry and Trigonometry at the Evelpides School (Central Military School) During the Kapodristrian Period . . . . . . . . . . . . . . . . 131 Andreas Kastanis The Overshadowing of Euclid’s Geometry by Legendre’s Géométrie in the Modern Greek Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Nikos Kastanis

vii

viii

Contents

Finite Element Methods with Higher Order Polynomials . . . . . . . . . . . . . . . . . . . 161 Konstantina C. Kyriakoudi and Michail A. Xenos On Local Asymptotics for Orthonormal Polynomials . . . . . . . . . . . . . . . . . . . . . . . 177 Eli Levin and D. S. Lubinsky New Trends in Geometric Function Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Khalida Inayat Noor and Mohsan Raza A Unified Approach to Extended General Quasi Variational Inclusions . . 237 Muhammad Aslam Noor, Khalida Inayat Noor, and Michael Th. Rassias On a Reverse Hilbert-Type Inequality in the Whole Plane with Multi-Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Michael Th. Rassias, Bicheng Yang, and Andrei Raigorodskii Generating Functions for the Fubini Type Polynomials and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Yilmaz Simsek and Neslihan Kilar Kleene Fixed Point Theorems and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Mihai Turinici On Ergodic States, Spontaneous Symmetry Breaking and Quasi-Averages 431 Walter F. Wreszinski and Valentin A. Zagrebnov Improvement of the Hardy Inequality and Legendre Polynomials . . . . . . . . . 459 Nikolaos B. Zographopoulos

On a Version of Jensen-Steffensen Inequality and a Note on Inequalities in Several Variables Shoshana Abramovich

1991 Mathematics Subject Classification. 26A51, 26B25, 26D15

1 Introduction We deal with special versions of Jensen-Steffensen inequality for convex functions, and we note on convex and subquadratic functions in one and several variables. Let I be an interval in .R, .f : I → R, be a convex function and .x = (x1 , . . . , xn ) be any non-negative n-tuple such that . n1 ωi = 1, then, the well known Jensen’s inequality f

 n 

.

i=1

 ωi xi

≤

n 

(1.1)

ωi f (xi )

i=1

holds (see [9, page 43]). Under more restrictive conditions we get Jensen-Steffensen inequality: Theorem 1 ([4]) Jensen-Steffensen’s inequality states that if .f : I →  R is convex, then (1.1) holds, where I is an interval in .R, x = x1,..., xn is any monotonic n-tuple in .I n , .ωi , .i = 1, . . . , n are real coefficients which are called Steffensen’s Coefficients, satisfying 0 ≤ Wj ≤ Wn ,

j = 1, . . . , n ,

Wn =

n 

ωi = 1 ,

(1.2)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_1

1

.

1

S. Abramovich () Department of Mathematics, University of Haifa, Haifa, Israel e-mail: [email protected]

2

S. Abramovich

Wj =

j 

ωi ,

Wj =

n 

ωi ,

j = 1, . . . , n.

i=j

i=1

In Sect. 2 we deal with variants of Jensen-Steffensen inequality and SlaterPeˇcari´c inequality, where in Theorems 8–11, we extend and refine the following Theorems 2–6: Theorem 2 ([4, Th2]) Let .f : I → R, where I is an interval in .R and let [a, b] ⊆ I, .a < b.Let .x = (x1 , . . . , xn ) be a monotonic n-tuple .x ∈ [a, b]n and .v = (v1 , . . . , vn ) , a real n-tuple such that .0 ≤ Vj ≤ Vn , .j = 1, . . . , n, .Vn > 0, j where .Vj = i=1 vi . If f is convex on I ,then .



n 1  a+b− vi xi Vn

f

.

 ≤ f (a) + f (b) −

i=1

n 1  vi f (xi ) . Vn i=1

Theorem 3 ([6, Theorem 1.2]) If f is a convex function on an interval containing an n-tuple .x = (x1 , · · · , xn ) such that .0< x1 ≤ x2 ≤ · · · ≤ xn and n .w = (w1 , · · · , wn ) is a positive n-tuple with . i=1 wi = 1, then  x1 + xn −

f

.

n 

 wi xi

≤ f (x1 ) + f (xn ) −

i=1

n 

wi f (xi ) .

i=1

Theorem 3 is a special case of Theorem 2. In Sect. 2 we use also a theorem proved in [8] by J. Peˇcari´c which is a refinement of Slater’s inequality that appears in [10]: Theorem 4 Under the same conditions leading to Jensen-Steffensen’s inequality, together with n  .

i=1

n ' i=1 ai xi f (xi ) ai f (xi ) /= 0 and M = n ∈ I, ' i=1 ai f (xi ) '

(1.3)

the inequality n  .

ai f (xi ) ≤ f (M)

(1.4)

i=1

 holds when . ni=1 ai = 1. Also, in Theorems 10 and 11 in Sect. 2 we refine Theorems 5 and 6 which appeared lately in [7, Theorem 2.1] and [7, Theorem 2.2]:

On Jensen-Steffensen and Other Inequalities

3

Theorem 5 If f is a convex function on an interval .[m, M], thenfor every x1 , . . . , xn ∈ [m, M] and every positive real numbers .w1 , . . . , wn with . ni=1 wi = 1  n  n    n n n     i=1 wi xi + xi −f wi xi ≤ wi f (xi ) , .f wi xi ≤ 2 wi f 2 .

i=1

i=1

i=1

i=1

and n  .

wi f (xi )−f

 n 

i=1



 ≤2

wi xi

i=1

n 

wi f (xi ) −

i=1

 n

n 

i=1 wi xi

wi f

+ xi

2

i=1

 .

Theorem 6 If f is a convex function on an interval .[m, M], thenfor every x1 , . . . , xn ∈ [m, M] and every positive real numbers .w1 , . . . , wn with . ni=1 wi = 1   n  . f M +m− wi xi

.

≤2

n 

 wi f

i=1

n

M +m−

i=1

≤ f (M) + f (m) −

n 

j =1 wj xj

+ xi

2



 −f

M +m−

n 

 wi xi

i=1

wi f (xi ) ,

i=1

and .

f (M) + f (m) − 

n 

 wi f (xi ) − f

M +m−

i=1

≤ 2 f (M) + f (m) −

n 

 wi xi

i=1

n  i=1

wi f (xi ) −

n  i=1

n   w x + x i i i wi f M + m − i=1 . 2

In Sect. 3 we note on convexity and subquadracity in several variables. We use there the same compound function in m variables .G1,m−1 (a1 , a2 , . . . , am ) as in Theorem 7: Theorem 7 ([2, Theorems 1 and 3]) Let .0 < fi ∈ B where B is the set of positive monotone functions defined on .x > 0, which are nonlinear on any subinterval of .x > 0 and which consists of the concave and convex functions. Then the

4

S. Abramovich

inequalities: n  .

  n n       G1,m−1 a1,k , a2,k , . . . , am,k ≥ G1 a1,k , G2,m−1 a2,k , . . . , am,k

k=1

.

≥ . . . ≥ G1,j

k=1

 n 

a1,k ,

k=1

.

n 

a2,k , . . . ,

k=1

≥ G1,m−1

n 

aj,k

k=1

 n 

a1,k ,

k=1

n 

k=1 n 

   Gj +1,m−1 a2,k , . . . , am,k

k=1

a2,k , . . . ,

k=1

n 

 am,k

k=1

hold for all .aik > 0, .i = 1, . . . , m, .k = 1, . . . , n if .fi (x), .i = 1, . . . , m − 2 are convex increasing, and .fm−1 (x)is convex, where .G1,m−1 (a1 , a2 , . . . , am ) is defined by (1.5), (1.6) and (1.7) as:  Gr (ar , ar+1 ) = ar f

.

ar+1 ar

,

1 ≤ r ≤ m − 1,

m ≥ 1,

(1.5)

and :

.

 Gr+1 ,r+2 (ar+2 , ar+3 ) = ar+1 fr+1

.

ar+2 fr+2 ar+1



ar+3 ar+2

= Gr+1 (ar+1 , Gr+2 ) , (1.6)

0 ≤ r ≤ m − 3, m ≥ 3,

.

and :

.

Gr+1,r+s (ar+1 , ar+2 , . . . , ar+s−1 , ar+s )

.

(1.7)

= Gr+1,r+s−1 (ar+1 , ar+2 , . . . , ar+s−1 , Gr+s (ar+s , ar+s+1 )) , 1 ≤ r + 1 ≤ r + s ≤ m, m ≥ 2.

.

In [3] and [5] the concept of superquadracity in one and several variables is discussed. Definition 1 ([5, Definition 1]) A function .f : (0, ∞)m → R is said to be superquadratic if for every .x ∈ (0, ∞)m there exists a vector .c(x) ∈ Rm such that f (y) ≥ f (x) + 〈c(x), y − x〉 + f (|y − x|)

.

(1.8)

holds for all .y ∈ (0, ∞)m . f is said to be strictly superquadratic if (1.8) is strict for all .x /= y, .0 < x.

On Jensen-Steffensen and Other Inequalities

5

The function f is said to be subquadratic if .−f is superquadratic. Lemma 1 ([5, Lemma3]) If .f : (0, ∞)m → R is a differentiable superquadratic function with .f (0) = 0 and .∇f (0) = 0, then for any .x ∈ (0, ∞)m .

〈∇f (x), x〉 − 2f (x) ≥ 0

(1.9)

We use Lemma 1 to show a set of functions that are convex but never superquadratic. We present also cases where these convex functions in two variables are subquadratic.

2 Variants of Jensen-Steffensen and Slater-Peˇcari´c Inequalities From Jensen-Steffensen’s inequality we get Theorems 8–10 for special cases of Jensen-Steffensen inequality: Theorem 8 Let .f : I → R, where I is an interval in .R and let .[a, b] ⊆ I, .a < b. Let .x = (x1 , . . . , xn ) be a monotonic n-tuple, .x ∈ [a,  b]n . Let .(w1 , w 2 , . . . wn ) = , α2 , .., αk , −βk+1 , . . . , −βm , γm+1 , . . . , γn ) where . ki=1 αi = 1, . m (α1 i=k+1 βi = 1, . ni=m+1 γi = 1, and each of .(α1 , α2 , .., αk ), .(βk+1 , . . . , βm ), .(γm+1 , . . . , γn ) are k Steffensen’s m coefficients as defined m in (1.2) in Theorem 1. Denoting .A = i=1 αi xi , .B = i=k+1 βi xi , .C = i=k+1 γi xi we get that when f is convex on I , the inequalities in k  .

αi f (xi ) − f (B) +

i=1

≥

n 

n 

(2.1)

γi f (xi )

i=m+1

wi f (xi )

i=1

=

k 

αi f (xi ) −

i=1

≥ f (A) −

m  i=k+1

m 

βi f (xi ) +

n 

βi f (xi ) + f (C) ≥ f

i=k+1

γi f (xi )

i=m+1

 A−

m 

 βi xi + C

i=k+1

= f (A − B + C) , hold. Moreover, even when .(x1 , . . . , xn ) is not a monotonic sequence, but .(x1 , . . . , xk ), .(xk+1 , . . . , xm ) and .(xm+1 , . . . , xn ) are increasing sequences and .(α1 , . . . , αk ) ,

6

S. Abramovich

(βk+1 , . . . , βm ) and .(γm+1 , . . . , γn ) are Steffensen’s coefficients but

.

xk+1 < xk ,

.

A ≤ xk+1 ,

xm > xm+1 ,

xm < C,

(2.2)

is satisfied, we get again that the inequalities in (2.1) hold. Proof With no loss of generality we assume that .(x1 , . . . , xn ) is an increasing sequence. The proof of the first inequality in (2.1) for the monotone n-tuple .(x1 , . . . , xn ) follows from Theorem 1 regarding the coefficients .βk+1 , . . . , βm and the terms .xk+1 , . . . , xm . The proof of the second inequality in (2.1) holds because the terms .αi , .i = 1, . . . , k, .βi , .i = k + 1, . . . , m and .γi , .i = m + 1, . . . , n are  Steffensen’s coefficients and therefore . ki=1 αi f (xi ) ≥ f (A) , .A ≤ xk ≤ xk+1 n and . i=m+1 γi f (xi ) ≥ f (C) , .C ≥ xm+1 ≥ xm . The .m + 2 − k coefficients .(1, −βk+1 , . . . , −βm , 1) are Steffensen’s coefficients for the increasing sequence .(A, xk+1 , . . . , xm , C) and therefore by Theorem 1 as well as Theorem 2 the third inequality in (2.1) holds. Also, it is obvious that (2.2) is sufficient for each of the inequalities in (2.1) to hold. This follows because the last inequality in (2.1), according to (2.2) the .(m − k + 2)-tuple .(A, xk+1 , . . . , xm , C) is increasing and the coefficients .(1, −βk+1 , . . . , −βm , 1) satisfy (2.2), the Jensen-Steffensen’s conditions and therefore (1.1) holds. This completes the proof of the theorem. ⨆ ⨅ It is easy to see that Theorem 8 can be generalized by extending it to .2N + 1 components instead of three components nif .x1 , . . . , xn and .ω1 , . . . , ωn satisfy Jensen-Steffensen’s theorem and besides . signs of the  1  2 n3 i=1 ωi = 1, also alternating n k ωi = 1, . ni=n ω = −1, . ω = 1, . . . , . sum . ni=1 i i i=n2 +1 i=nk−1 +1 ωi = −1, 1 +1 n . ω = 1. i=nk +1 i Example 1 In this example we demonstrate that in cases satisfied by condition (2.2) the inequality 8  .

i=1

 ηi f (xi ) ≥ f

8  i=1

 ηi xi ,

8 

ηi = 1

i=1

holds for convex functions f , although .(y1 , . . . , y8 ) below is not a monotone permutation of the following .(x1 , . . . , x8 ) and .(η1 , . . . , η8 ) are not Steffensen’s coefficients. Let the increasing .xi and its coefficients .ηi , .i = 1, . . . , 8 which do not satisfy (1.2) be: .

x1 = 1 x2 = 2.5 x3 = 3 x4 = 3.5 x5 = 4 x6 = 5 x7 = 6 x8 = 7 η1 = 1 η2 = −0.5 η3 = −1 η4 = −0.5 η5 = 1 η6 = 1/3 η7 = 1/3 η8 = 1/3.

On Jensen-Steffensen and Other Inequalities

7

  In this case . 8i=1 ηi = 1 and . 8i=1 ηi xi = 6. We see that for .f (x) = x 2 we get: 53, 25 =

8 

.

ηi xi2 ≥

 8 

i=1

2 = 36.

ηi xi

i=1

The reason is that when rearranging the couple .(xi , ηi ), to .(yi , ωi ),.i = 1, . . . , 8 into .

y1 = 1 y2 = 3 y3 = 4 y4 = 2.5 y5 = 3.5 y6 = 5 y7 = 6 y8 = 7 , ω1 = 1 ω2 = −1 ω3 = 1 ω4 = −0.5 ω5 = −0.5 ω6 = 1/3 ω7 = 1/3 ω8 = 1/3

  we get that . 3i=1 ωi yi = 2 = A < 2.5 = y4 , and . 8i=6 ωi yi = 6 = C > 3.5 = y5 , which means that (2.2) is satisfied. Therefore, 8  .

ηi xi =

8 

i=1

ωi yi = 2 +

i=1

5 

ωi yi + 6,

i=4

and 8  .

f (ηi xi ) =

i=1

8 

f (ωi yi ) = f (2) +

i=1

5 

f (ωi yi ) + f (6) .

i=4

Hence, as the four touple .(2, 2.5, 3, 5, 6) with the coefficients .(1, −0.5, −0.5, 1) satisfy the conditions of Jensen-Steffensen inequality we get that 8 

ηi f (xi )

.

i=1

=

8 

ωi f (yi ) = f (2) +

i=1



≥f

=f

2+  8 

5  i=4



5  i=4

ωi yi + 6 = f 

ωi f (yi ) + f (6)  8 

 ωi yi

i=1

ηi xi .

i=1

We prove now a theorem similar to Theorem 8 by combining Jensen-Steffensen and Slater-Peˇcari´c inequalities and we obtain a new inequality:

8

S. Abramovich

Theorem 9Let .(w1 , w2 , . . . wn ) = (α1 , α2 , .., αk , −βk+1 , . . . , −βm , γm+1 , . . . , n γn ) where . ki=1 αi = 1, . m β = 1, . i i=k+1 i=m+1 γi = 1, and each .(α1 , α2 , .., αk ), .(βk+1 , . . . , βm ), .(γm+1 , . . . , γn ) are Steffensen’s coefficients. Denoting .A = m ' k m m i=k+1 βi xi f (xi ) ∈ ' i=1 αi xi , .B = i=k+1 βi xi , .C = i=k+1 γi xi , and let .M = m .

i=k+1 βi f

[xk+1 , xm ], we get that the inequalities k  .

αi f (xi ) − f (B) +

i=1

≥

n 

n 

(xi )

γi f (xi )

i=m+1

wi f (xi )

i=1

=

k 

αi f (xi ) −

i=1

≥ f (A) − f

m  i=k+1

 m

n 

βi f (xi ) +

γi f (xi )

i=m+1 '

βi xi f (xi ) i=k+1 ' m i=k+1 βi f (xi )



+ f (C) ≥ f (A − M + C) ,

hold when .(x1 , . . . , xn ) is a monotonic n-tuple and f is convex on an interval I such that .xi ∈ I , .i = 1, . . . , n. Proof As before, we prove the theorem assuming that .(x1 , . . . , xn ) is an increasing n-tuple. The first inequality is the same as in Theorem 8. The second inequality follows from Slater-Peˇcari´c inequality. The third inequality follows from Jensen-Steffensen inequality as it is given that .M ∈ .[xk+1 , xm ]. Therefore, .(A, M, C) is a increasing series, .(1, −1, 1) are coefficients satisfying Jensen-Steffensen’s conditions and f is a convex function. The proof of the theorem is complete. ⨆ ⨅ ˇ The inequality in Lemma 2, called Cebyšev inequality, is used in Remark 1: Lemma 2 ([9, Page 197]) Let .x and .y be two .n-tuple of real numbers monotonic in the same direction, and let .β be a positive n-tuple and . ni=1 βi = 1. Then, n  .

i=1

βi xi yi ≥

n  i=1

βi xi

n 

(2.3)

βi yi .

i=1

Remark 1 Comparing the lower bound .f (A − B + C) obtained in Theorem 8 with the bound .f (A − M + C) obtained in Theorem 9 we see that in the special case when f is increasing, convex and .βi > 0, .i = k + 1, . . . , m, the inequality ' .f (A − M + C) ≤ f (A − B + C) holds. This is because by choosing .yi = f (xi ) m ' m i=k+1 βi xi f (xi ) ˇ in (2.3) we get that .B = = M by Cebyšev ' i=k+1 βi xi ≤ m i=k+1 βi f

(xi )

inequality. It means that the lower bound obtained by using Jensen-Steffensen’s

On Jensen-Steffensen and Other Inequalities

9

theorem is better in this case than the upper bound obtained by using Slater-Peˇcari´c inequality. Theorem 5 is a special case of: Theorem 10 Let f be a convex function on an interval .[a, b], let .xi ∈ [a, b], .i = 1, . . . , n be an increasing  sequence, .wi , . i = i, . . . , n, satisfy (1.2), the Steffensen’s conditions. Then, for .x = n1 wi xi , and . n1 wi = 1, and .0 ≤ t ≤ 1 the inequality tf (x) ≤

n 

.

wi f ((1 − t) x + txi ) − (1 − t) f (x)

(2.4)

i=1

holds. Also, if .wi ≥ 0 then tf (x) ≤

n 

.

wi f ((1 − t) x + txi ) − (1 − t) f (x) ≤ t

i=1

n 

wi f (xi )

(2.5)

i=1

holds too. Proof Inequality (2.4) is an immediate result of Theorem 1, that is JensenSteffensen’s inequality. The first inequality in (2.5) is Jensen inequality (1.1) for the coefficients .wi ≥ 0, .i = 1, . . . n. To prove the second inequality in (2.5) we use Jensen’s inequality for the two coefficients .0 ≤ t ≤ 1 and .(1 − t), from which we get n  .

wi f ((1 − t) x + txi ) ≤

i=1

.

= (1 − t) f (x) +

n 

wi ((1 − t) f (x) + tf (xi ))

i=1 n 

wi tf (xi ) .

i=1

Hence (2.5) holds. The proof of the theorem is complete.

⨆ ⨅

It is easy to see that Theorem 6 is a special case of Theorem 5 when we consider these theorems as consequences of Theorem 1. Theorem 11 is a refinement and extension of Theorems 5 and 6. Therefore, we prove in Theorem 11 a generalization of Theorem 6 when choosing in Theorem 11 .t = 12 . Theorem 11 Let f be a convex function on .I = [A, C] .∈ R and let (A, x1 , x2 , . . . , xm , C) be an increasing .(m + 2)-tuple. Let .βi , .i = 1, . . . , m be

.

10

S. Abramovich

Jensen-Steffensen’s coefficients. Then the following inequality for .0 ≤ t ≤ 1  A−

tf

.

m 

 βi xi + C

i=1

≤

m 

⎛

(2.6)

⎛

βi f ⎝A − ⎝txi + (1 − t)

i=1

m 

⎞

⎞

βj xj ⎠ + C ⎠

j =1

⎛

− (1 − t) f ⎝A −

m 

⎞

βj xj + C ⎠

j =1

holds. If in addition, .βj ≥ 0, .j = 1, . . . , m then the inequalities m 

t

.

⎛ βi f ⎝A −

=

m 

⎛ ⎛ βi ⎝f ⎝A −

i=1

≤

⎞ βj xj + C ⎠

(2.7)

j =1

i=1

.

m 

m 

m 

βj xj + C ⎠ − (1 − t) f ⎝A −

j =1

⎛

⎛

βi ⎝(1 − t) f ⎝A −

i=1

m 

− (1 − t) f ⎝A −

m 

m 

⎞⎞ βj xj + C ⎠⎠

j =1

⎞

⎞

βj xj + C ⎠ + tf (A − xi + C)⎠

j =1

⎛

⎛

⎞

⎞

βj xj + C ⎠

j =1

.

=t

m 

βi f (A − xi + C) ≤ t

i=1

 .

= t f (A) −

βi (f (A) − f (xi ) + f (C))

i=1 m  i=1

hold.

m 





βi f (xi ) + f (C) ≤ t f (A) − f



m  i=1

 βi xi

 + f (C)

On Jensen-Steffensen and Other Inequalities

11

Proof To prove (2.6) we see that for Steffensen’s coefficients .βi , .i = 1, . . . , m the inequality m  .

⎛

⎛

βi f ⎝A − ⎝txi + (1 − t)

m 

⎞

βj xj ⎠ + C ⎠

j =1

i=1

 ≥ tf

A−

m 

⎞

⎛



βi xi + C + (1 − t) f ⎝A −

m 

⎞ βj xj + C ⎠

j =1

i=1

 holds because the sequence of m terms .A + txi + (1 − t) m j =1 βj xj + C, .i = 1, . . . , m is increasing.  To prove (2.7) where .βi > 0, .i = 1, . . . , m and . m i=1 βi = 1, we see that: The first inequality in (2.7) is Jensen’s inequality for .n = 2 because .0 ≤ t ≤ 1. The second inequality in (2.7) follows from Theorem 2 for the 3-tuple .(A, xi , C). The last inequality in (2.7) is Jensen’s inequlity because .βi > 0, .i = 1, . . . , m and  m . ⨆ ⨅ i=1 βi = 1.

3 Remarks on Convexity and Subquadracity in Several Variables In Theorem 12 we deal with a set of functions .G (x, y) : R2+ → R+ which are convex but never superquadratic: Theorem 12 Let .f : R+ → R+ be a differentiable convex function. Then the function .G (x, y) = yf xy where .(x, y) > (0, 0), .G (0, 0) = 0, .∇G (0, 0) = (0, 0), is convex but not superquadratic in the two variables .(x, y) ∈ (0, ∞)2 . Proof First we show that .G (x, y) is convex:  Let .ti ≥ 0, .i = 1, . . . , n, .(xi , yi ) ∈ (0, ∞)2 , . ni=1 ti = 1, then because f is convex on .x ≥ 0 n  .

i=1

ti G (xi , yi ) =

n 

 ti yi f

i=1

=G

 n  i=1

xi yi

ti xi ,

≥

n 

n  i=1



 n i=1 ti xi  ti yi f n i=1 ti yi

ti yi

i=1

which means that .G (x, y) is convex. Now we show that .G (x, y) cannot be superquadratic.

12

S. Abramovich

If a non-negative function in two variables is superquadratic then according to Lemma 1, it satisfies the inequality .

〈▽G (x, y) , (x, y)〉 ≥ 2G (x, y) .

  We show that for our type of .G (x, y) where .G (x, y) = yf xy this inequality is reversed because      x x ' x x ' f − f , (x, y) ,f . 〈▽G (x, y) , (x, y)〉 = y y y y .

= yf

 x = G (x, y) ≤ 2G (x, y) . y

  Hence in this case when f is non-negative .G (x, y) = yf xy can not be superquadratic in two variables. The proof of the theorem is complete. ⨆ ⨅   An example of functions of the type .G (x, y) = yf xy is: .G (x, y) =   p  p1 , .G (0, 0) = 0, .(x, y) ≥ (0, 0), .p > 1. These functions are convex y 1 + xy and for .p > 1 are not superquadratic as proved in Theorem 12, but are subquadratic as proved in [5, Example 3]. Moreover, in this example in [5] it is shown that the n p  p1 , .G (0, . . . , 0) = 0, .(x1 , . . . , xn ) ≥ 0 are functions .G (x1 , . . . , xn ) .= i=1 xi subquadratic on .Rn+ for .p > 1. In order to generalize the convexity part of Theorem 12 and to extend Theorem 7 we use the definition of .G1,m−1 (a1 , a2 , . . . , am ) as in (1.5), (1.6) and (1.7). The proof is similar to the proofs of [2, Theorem 1 and Theorem 3]. Theorem 13 Let .0 < fi ∈ B where B is the set of positive monotone functions defined on .x > 0, which are non-linear on any subinterval of .x > 0, which consists of the concave and convex functions, then .G1,m−1 (a1 , a2 , . . . , am ) is convex, and the inequalities in n  .

  tk G1,m−1 a1,k , a2,k , . . . , am,k

k=1

≥ G1

 n 

tk a1,k ,

k=1

≥ . . . ≥ G1,j

n 

(3.1)

   G2,m−1 a2,k , . . . , am,k

k=1

 n  k=1

tk a1,k ,

n  k=1

tk a2,k , . . . ,

n  k=1

tk aj,k

n  k=1

   Gj +1,m−1 a2,k , . . ., am,k

On Jensen-Steffensen and Other Inequalities

≥ G1,m−1

 n  k=1

tk a1,k ,

n  k=1

tk a2,k , . . . ,

13 n 

 tk am,k

k=1

 hold for all .aik > 0, .i = 1, . . . , m, .k = 1, . . . , n, .0 ≤ tk ≤ 1, . nk=1 tk = 1 if .fi (x), .i = 1, . . . , m − 1 are ordered as follows

.

⎫ fj2k +1 , . . . , fj2k+1 are convex functions ⎬ fj2k+1 +1 , . . . , fj2k+2 are concave functions ⎭ j0 = 0, j2k0 +2 ≤ m − 1 k = 0, . . . , k0

(3.2)

the fjk are decreasing and all the otherfi (x) are increasing. Proof First we show that under the simpler conditions of Theorem 7 the inequalities in (3.1) hold, that is, we show that (3.1) holds for convex increasing functions .fi ∈ B, .i = 1, . . . , m − 2 and for a convex .fm−1 (x). The first inequality in (3.1) holds because .f1 is a convex function. The second inequality in (3.1) holds because .fj is convex and .fi (x), .i = 1, . . . , j − 1, are increasing. The last inequality holds because .fm−1 (x) is convex and .fi (x), .i = 1, . . . , m − 2 are increasing. Thus in this case (3.1) is proved. We now see that inequalities in (3.1) hold under the conditions in (3.2) on .fi , .i = 0, 1, . . . m − 1. The index .j = 0 means that .f1 (x) is convex. The .fjk th being decreasing means that before the concave function .fj2k+1 +1 , there is a convex decreasing function .fj2k+1 , and before the convex function .fj2k +1 , .k ≥ 1 there is a concave decreasing function .fj2k . From these it is easy to see that if (3.1) holds then (3.2) holds. ⨆ ⨅ Remark 2 It is obvious that similar to Theorem 13 we get also that when .fi , .i = 1, . . . , m − 2 are increasing and concave and .fm−1 is concave then the reverse of the inequalities in (3.1) hold. We finish with an example, this time of a concave function in several variables that involves a function F on .x > 0 for which .F (x) and .xF x1 are increasing, like the case where .F (x) = x p , .x > 0, .0 < p < 1. This example is similar to [1, Example 3.1].   Example 2 Let .F (x) be positive concave functions, .F (x) and .xF x1 be increasing functions on .(0, ∞) , and let .fi , and = 1, . . ., m −  .gi be positive functions  .i  a3 a2 am 1. Let .G1,m−1 (a1 , . . . , am ) = a1 f1 a1 f2 a2 , . . . , fm−1 am−1 , . . . , and      b3 b2 bm , . . . , .G1,m−1 (b1 , . . . , bm ) = b1 g1 g , . . . , g satisfy the con2 m−1 b1 b2 bm−1 ditions of Remark 2.

14

S. Abramovich

Then .



  G1,m−1 b1,k , b2,k , . . . , bm,k   (3.3) a G , a , . . . , a 1,m−1 1,k 2,k m,k k=1  n   n    k=1 tk G1,m−1 b1,k , b2,k , . . . , bm,k   ≤ tk G1,m−1 a1,k , a2,k , . . . , am,k F n k=1 tk G1,m−1 a1,k , a2,k , . . . , am,k k=1  n  n n    ≤ G1,m−1 tk a1,k , tk a2,k , . . . , tk am,k n 

  tk G1,m−1 a1,k , a2,k , . . . , am,k F

 ×F

k=1

k=1

k=1

n  n n G1,m−1 k=1 tk b1,k , k=1 tk b2,k , . . . , k=1 tk bm,k n  , n n G1,m−1 k=1 tk a1,k , k=1 tk a2,k , . . . , k=1 tk am,k

 for all .ai,k > 0, .bi,k > 0, .i = 1, . . . , m, .k = 1, . . . , n, .0 ≤ tk ≤ 1, . nk=1 tk = 1. Indeed, from the concavity of F , .fi and .gi , .i = 1, . . . , m − 1 and because .fi and .gi , .i = 1, . . . , m − 2, are increasing, we get by Jensen’s inequality that

.



  G1,m−1 b1,k , b2,k , . . . , bm,k   (3.4) G1,m−1 a1,k , a2,k , . . . , am,k k=1  n   n    k=1 tk G1,m−1 b1,k , b2,k , . . ., bm,k   , ≤ tk G1,m−1 a1,k , a2,k , . . ., am,k F n k=1 tk G1,m−1 a1,k , a2,k , . . ., am,k k=1 n 

  tk G1,m−1 a1,k , a2,k , . . . , am,k F

n  .

  tk G1,m−1 a1,k , a2,k , . . . , am,k

k=1

≤ G1,m−1

 n 

tk a1,k ,

k=1

n 

tk a2,k , . . . ,

k=1

n 

 (3.5)

tk am,k

k=1

and n  .

  tk G1,m−1 b1,k , b2,k , . . . , bm,k

k=1

≤ G1,m−1

 n  k=1

tk b1,k ,

n  k=1

tk b2,k , . . . ,

n 

 tk bm,k

(3.6)

k=1

  hold, and as .F (x) and .xF

1 x

are increasing we get from (3.4), (3.5) and (3.6) that

(3.3) holds, and we get in this example a concave function in .R2m .

On Jensen-Steffensen and Other Inequalities

15

References 1. S. Abramovich, Convexity, subadditivity and generalized Jensen inequality. Ann. Funct. Anal. 4(2), 183–194 (2013) 2. S. Abramovich, J.E. Pecaric, Convex and concave functions and generalized holder inequalities. Soochow J. Math. Taiwan 14, 261–272 (1998) 3. S. Abramovich, G. Jameson, G. Sinnamon, Refining Jensen’s inequality. Bull. Math. Soc. Sci. Math. Roumanie (Novel Series) 47(95), 3–14 (2004) 4. S. Abramovich, M.K. Bakula, M. Mati´c, J. Peˇcari´c, A variant of Jensen-Steffensen’s inequality and quasi-arithmetic means. J. Math. Anal. Appl. 307, 370–386 (2005) 5. S. Abramovich, S. Bani´c, M. Mati´c, Superquadratic functions in several variables. J. Math. Anal. Appl. 327, 1444–1460 (2007) 6. A.McD. Mercer, A variant of Jensen’s inequality. J. Inequal. Pure Appl Math. 4(4), Article 73 (2003) 7. L. Nasiri, A. Zardadi, H.R. Moradi, Refining and reversing Jensen’s inequality. Oper. Matrices 16(1), 19–27 (2022) 8. J. Peˇcari´c, A companion inequality to jensen-steffensen’s inequality. J. Approx. Theory 44, 289–291 (1985) 9. J. Peˇcari´c, F. Proschan, Y.L.Tong, Convex functions, partial ordering and statistical applications (Academic Press, Cambridge, 1992) 10. M.L. Slater, A companion inequality to Jensen’s inequality. J. Approx. Theory 32, 160–166 (1981)

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law Oanh Chau, Adrien Petrov, and Arnaud Heibig

Mathematics Subject Classification (2000) 74M15, 74M10, 74F05, 74H20, 74H25, 34G25

1 Introduction Since the dawn of time, situations of contact between deformable bodies abound in everyday life and engineering applications, where the numerous forces acting on the system may lead to the appearance of microcracks, and deteriorate the mechanical equipments. It is then important to understand the complexity of the contact phenomena, in order to guarantee the safety of the mechanism of the system. Therefore considerable efforts have been achieved in modeling, mathematical analysis and numerical simulations, within the weak distributional formulation framework, expressed in terms of evolutional variational inequalities and hemivariational inequalities. The literature dedicated to this field is increasing day by day. The state of the art can be found in the masterpieces [4, 6, 7]. This work is a continuation of the paper in [2], where the authors studied a frictional unilateral contact problem undergoing the thermal expansion and damage, for viscoelastic solid. The friction is modeled by the Coulomb’s dry friction law. Here we investigate a thermal contact with friction, for time depending long memory visco-elastic materials, with or without the clamped condition. The contact conditions are unilateral. The friction is modeled by an inequality of sub-differential type.

O. Chau () University of Reunion Island, PIMENT, Saint-Denis Messag Cedex, Reunion Island, France e-mail: [email protected] A. Petrov · A. Heibig Université de Lyon, Villeurbanne, France e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_2

17

18

O. Chau et al.

The paper is organized as follows. In Sect. 2 we describe the mechanical problem, its corresponding variational formulation, and then we claim the main existence and uniqueness result under specific assumptions, that we prove in Sect. 3.

2 The Contact Problem In this section we describe the mechanical problem, list the assumptions on the data and derive the corresponding variational formulation. Then we state an existence and uniqueness result on displacement field and temperature, which we will prove in the next section. The physical setting is as follows. A visco-elastic body occupies a domain .Ω in d .R (.d = 1, .d = 2 or .d = 3) with a Lipschitz boundary .Γ that is partionned into three disjoint measurable parts, .Γ1 , .Γ2 and .Γ3 . We denote by .ν the unit outward normal on .Γ . Let .[0, T ] be the time interval of interest, where .T > 0. The body is clamped on .Γ1 × (0, T ) and therefore the displacement field vanishes there. Here we suppose that meas.(Γ1 ) = 0 or meas.(Γ1 ) > 0, which means that .Γ1 may be an empty set or reduced to a finite set of points. We assume that a volume force of density .f0 acts in .Ω × (0, T ) and that surface tractions of density .f2 act on .Γ2 × (0, T ). The body may come in contact with an obstacle, the foundation, over the contact surface .Γ3 . The contact is unilateral involving here both the normal displacement, the normal velocity and the normal component of the Cauchy stresss vector. The model of the friction is specified by a general sub-differential condition, where thermal effects may occur in the frictional contact with the basis. We are interested in the dynamic evolution of the body. Let us recall now some classical notations, see e.g. [4] for further details. We denote by .Sd the space of second order symmetric tensors on .Rd , while “ .· ” and .| · | will represent the inner product and the Euclidean norm on .Sd and .Rd . Everywhere in the sequel the indices i and j run from 1 to d, summation over repeated indices is implied and the index that follows a comma represents the partial derivative with respect to the corresponding component of the independent variable. We also use the following notation:  d H = L2 (Ω) ,

.

H = { σ = (σij ) | σij = σj i ∈ L2 (Ω), 1 ≤ i, j ≤ d},

H1 = { u ∈ H | ε(u) ∈ H },

.

H1 = { σ ∈ H | Div σ ∈ H }.

Here .ε : H1 −→ H and .Div : H1 −→ H are the deformation and the divergence operators, respectively, defined by: ε(u) = (εij (u)),

.

εij (u) =

1 (ui,j + uj,i ), 2

Div σ = (σij,j ).

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

19

The spaces H , .H , .H1 and .H1 are real Hilbert spaces endowed with the canonical inner products given by:  (u, v)H =

 (σ, τ )H =

ui vi dx,

.

Ω

(u, v)H1 = (u, v)H + (ε(u), ε(v))H ,

σij τij dx, Ω

(σ, τ )H1 = (σ, τ )H + (Div σ, Div τ )H .

.

Recall that .D(Ω) denotes the set of infinitely differentiable real functions with compact support in .Ω; and .W m,p (Ω), .H m (Ω) := W m,2 (Ω), .m ∈ N, .1 ≤ p ≤ +∞ for the classical real Sobolev spaces; .Lp (U ; X) the classical .Lp spaces defined on U with values in X. To continue, the mechanical problem is then formulated as follows. Problem Q Find a displacement field .u : (0, T ) × Ω −→ Rd and a stress field . σ : (0, T ) × Ω −→ Sd and a temperature field .θ : (0, T ) × Ω −→ R+ such that for a.e. .t ∈ (0, T ):

.

σ (t) = A (t)ε(u(t)) ˙ + G (t)ε(u(t)) t

+

B(t − s) ε(u(s)) ds + Ce (t, θ (t)) in Ω

(1)

u(t) ¨ = Div σ (t) + f0 (t) in

(2)

0 .

u(t) = 0 on

.

σ (t)ν = f2 (t) on

uν (t) ≤ 0,

σν (t) ≤ 0,

(3)

Γ1

.

.

Ω

(4)

Γ2

σν (t) · u˙ ν (t) = 0

on

Γ3 ,

(5)

uν (t) = 0 =⇒ ϕc (t, wτ )−ϕc (t, u˙ τ (t)) ≥ − στ (t)·(wτ − u˙ τ (t)) ∀w ∈ Uad on Γ3 (6) .θ˙ (t) − div(Kc (t, ∇θ (t))) = De (t, u(t)) ˙ + q(t) in Ω, (7)

.

.

' − Kc (t, x, ∇θ (t, x)) ν := ϕthermal (t, x, θ (t, x))

θ (t) = 0 on

.

θ (0) = θ0

.

u(0) = u0 ,

.

a.e.

Γ1 ∪ Γ2 in Ω

u(0) ˙ = v0

x ∈ Γ3 ,

(8) (9) (10)

in Ω

(11)

Here, (1) is the Kelving Voigt’s time-dependent long memory thermo-viscoelastic constitutive law of the body, where .σ represents the stress tensor; .A denotes

20

O. Chau et al.

the viscosity operator depending on the velocity of infinitesimal deformations .ε(u), ˙ with the notation : for .τ ∈ Sd , .A (t)τ = A (t, ·, τ ) some function defined on .Ω; here a dot above a quantity represents the derivative of the quantity with respect to the time variable; .G is the elastic operator depending on the linearized strain tensor .ε(u) of infinitesimal deformations, with .G (t)τ = G (t, ·, τ ) which is defined on .Ω. The term .B(t)τ = B(t, ·, τ ) represents the so called relaxation tensor which is time-depending on the linearized strain tensor and is defined on .Ω. Recall that the visco-elastic short memory corresponds to the case .B ≡ 0. The last tensor .Ce (t, θ ) := Ce (t, ·, θ ) denotes the thermal expansion tensor depending on time and on the temperature, defined on .Ω. For example, Ce (t, θ ) := −θ Cexp (t) in Ω,

.

where Cexp (t) := (cij (t, ·))

.

is some time-depending expansion tensor, defined on .Ω. In (2) is the dynamic equation of motion where the mass density .ϱ ≡ 1. The equation in (3) is the clamped condition. In (4) is the traction condition, where .σ ν represents the Cauchy stress vector. On the contact surface, the general relations in (5) are the so-called Signorini’s boundary conditions (see [8]), involving here both the normal displacement .uν , the normal velocity .u˙ ν and the normal component of the Cauchy stresss vector .σν . Recall that the condition .uν ≤ 0 represents a non penetration of the surface asperities into the obstacle : the inequality .uν < 0 means that there is no contact of the surface with the obstacle, and the equality .uν = 0 denotes contact. The last condition stipulates that the product .σν u˙ ν vanishes almost everywhere on .(0, T ) × Γ3 . It’s physical meaning can be seen as a natural consequence of the non penetrability, under some regularity assumption. To show that, let fix .(t, x) ∈ (0, T ) × Γ3 . Case 1 : .uν (t, x) < 0. There is no contact, then .σν (t, x) = 0 and .σν u˙ ν = 0 at .(t, x). Case 2 : .uν (t, x) = 0. Consider three sub-cases : (i) .u˙ ν (t, x) > 0; (ii) .u˙ ν (t, x) = 0; (iii) .u˙ ν (t, x) < 0. ν (t,x) In the case (i), as . uν (t+h,x)−u → u˙ ν (t, x), .h → 0, .h > 0; then .uν (t +h, x) > h 0 for .h > 0 small enought, which contradicts the non penetration. In the case (ii), we have .σν u˙ ν = 0 at .(t, x). In the case (iii), as in (i) we deduce that .uν (t + h, x) < 0 for any .h > 0 small enought, thus there is no contact and .σν (t + h, x) = 0 for any .h > 0 small enought, and then the continuity regularity implies that .σν (t, x) = 0 and .σν u˙ ν = 0 at .(t, x). The friction on the contact surface is modeled by the equation in (6), which is a sub-differential boundary condition, in the classical framework of convex analysis.

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

21

Here, .Uad denotes the space of admissible displacements defined in the case of unilateral contact by Uad = {w : Ω −→ Rd , wν ≤ 0 on Γ3 }.

.

For bilateral contact, we take Uad = {w : Ω −→ Rd , wν = 0 on Γ3 }.

.

To continue, .σ τ represents the tangential component of the Cauchy stress vector; and .ϕc : (0, T )×Γ3 ×Rd −→ R is a given sub-differential friction contact function. Various situations may be modelled by such a condition, see below at the end of this Section. The differential equation (7) describes the evolution of the temperature field, where .Kc (t, ∇θ ) := Kc (t, ·, ∇θ ) is some nonlinear thermal conductivity function defined on .Ω, depending on time and on the temperature gradient .∇θ . For example, denote by Kc (t, ·) := (kij (t, ·))

.

the thermal conductivity tensor defined on .Ω, we could consider Kc (t, ·, ∇θ ) = Kc (t, ·) ∇θ.

.

˙ := De (t, ·, u(t)) ˙ represents some viscosityIn the second member, .De (t, u(t)) deformation heat nonlinear function defied on .Ω and depending on the displacement velocity, whereas .q(t) denotes the density of volume heat sources. For example, De (t, u(t)) ˙ = −Cexp (t) :

.

▽

u(t) ˙ = −cij (t, ·)

∂ u˙ i (t). ∂ xj

The associated temperature boundary condition is given by (8) and (9), where ϕthermal are some thermal boundary function defined on .(0, T ) × Γ3 × R. Here

.

' ϕthermal (t, x, r) := [ϕthermal (t, x, ·)]' (r), ∀(t, x, r) ∈ (0, T ) × Γ3 × R

.

denotes the derivative on the third variable of .ϕthermal . Taking the previous example for .Kc , we have Kc (t, x, ∇θ ) ν = kij (t, x)

.

∂θ νi . ∂ xj

Let consider the following standard example ϕthermal (t, x, r) :=

.

1 ke (t, x)(r −θR (t, x))2 , ∀(t, x, r) ∈ (0, T )×Γ3 ×R, 2

(12)

22

O. Chau et al.

where .θR is the temperature of the foundation, and .ke is the heat exchange coefficient between the body and the obstacle. We obtain ' ϕthermal (t, x, r) = ke (t, x) (r − θR (t, x)), (t, x, r) ∈ (0, T ) × Γ3 × R.

.

Finally in (10) and (11), .θ0 , u0 , v0 represent the initial temperature, displacement and velocity respectively. One may remark that since .ϕc is assumed real-valued, then unilateral contact, defined by indicator functions taking infinite values, is excluded. So the body is in fixed contact with the foundation of the body according to a friction law. This is consistent with the linear heat conduction modeled in (7). We insist that the new feature here is that we may have the absence of the usual claimed condition in the case where meas.(Γ1 ) = 0. However, there is coerciveness with regard to the temperature by (8). To derive the variational formulation of the mechanical problems (1)–(11) we need additional notations. Let consider the following space V of admissible displacements, that we assume to be a closed subspace of .H1 in the sequel : V = { w ∈ H1 | w = 0 on Γ1 } ∩ Uad .

.

On V we consider the inner product given by (u, v)V = (ε(u), ε(v))H + (u, v)H

∀ u, v ∈ V ,

.

and let .‖ · ‖V be the associated norm, i.e. ‖v‖2V = ‖ε(v)‖2H + ‖v‖2H

.

∀ v ∈ V.

It follows that .‖·‖H1 and .‖·‖V are equivalent norms on V and therefore .(V , ‖·‖V ) is a real Hilbert space. Moreover, by the Sobolev’s trace theorem, we have a constant .C0 > 0 depending only on .Ω, and .Γ3 such that ‖v‖L2 (Γ3 ) ≤ C0 ‖v‖V

.

∀ v ∈ V.

Consider then the following spaces for the temperature field: E = {η ∈ H 1 (Ω), η = 0 on

.

Γ1 ∪ Γ2 };

F = L2 (Ω).

The spaces E and F , endowed with their respective canonical inner product, are Hilbert spaces.

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

23

Identifying then H and F with their own duals, we obtain two Gelfand evolution triples (see e.g. [9] II/A p. 416): V ⊂ H ≡ H ' ⊂ V ',

.

E ⊂ F ≡ F ' ⊂ E'

where the inclusions are continuous and dense. Finally, we use the notation .〈·, ·〉V ' ×V and .〈·, ·〉E ' ×E to represent the duality pairing between .V ' and V , and respectively between .E ' and E, which means : 〈u, v〉V ' ×V = 〈u, v〉H , ∀ u ∈ H, ∀ v ∈ V .

.

and 〈η, ξ 〉E ' ×E = 〈η, ξ 〉F , ∀ η ∈ F, ∀ ξ ∈ E.

.

In the study of the mechanical problem (1)–(11), we assume that the viscosity operator .A : (0, T ) × Ω × Sd −→ Sd , .(t, x, τ ) − I → A (t, x, τ ) satisfies ⎧ (i) A (·, ·, τ ) is measurable on (0, T ) × Ω, ∀τ ∈ Sd ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ii) A (t, x, ·) is continuous on Sd for a.e. (t, x) ∈ (0, T ) × Ω; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (iii) there exists mA > 0 such that (A (t, x, τ1 ) − A (t, x, τ2 )) · (τ1 − τ2 ) ≥ mA |τ1 − τ2 |2 , . ⎪ ⎪ ∀τ1 , τ2 ∈ Sd , for a.e. (t, x) ∈ (0, T ) × Ω; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (iv) there exists c0A ∈ L2 ((0, T ) × Ω; R+ ), c1A > 0 such that ⎪ ⎪ ⎪ ⎪ |A (t, x, τ )| ≤ c0A (t, x) + c1A |τ |, ⎪ ⎪ ⎩ ∀τ ∈ Sd , for a.e. (t, x) ∈ (0, T ) × Ω.

(13)

In this paper for every .t ∈ (0, T ), .τ ∈ Sd we denote by .A (t) = A (t, ·, ·) a functional which is defined on .Ω × Sd and .A (t) τ = A (t, ·, τ ) some function defined on .Ω. The elasticity operator .G : (0, T ) × Ω × Sd −→ Sd satisfies: ⎧ (i) there exists LG > 0 such that ⎪ ⎪ ⎪ ⎪ |G (t, x, ε1 ) − G (t, x, ε2 )| ≤ LG |ε1 − ε2 | ⎪ ⎪ ⎨ ∀ε1 , ε2 ∈ Sd , a.e. (t, x) ∈ (0, T ) × Ω ; . ⎪ ⎪ ⎪ (ii) G (·, ·, ε) is Lebesgue measurable on (0, T ) × Ω, ∀ε ∈ Sd ; ⎪ ⎪ ⎪ ⎩ (iii) the mapping G (·, ·, 0) ∈ H .

(14)

We put again .G (t)τ = G (t, ·, τ ) some function defined on .Ω for every .t ∈ (0, T ), τ ∈ Sd .

.

24

O. Chau et al.

The relaxation tensor .B (Bij kh (t, x) τkh ) satisfies

:

(0, T ) × Ω × Sd −→ Sd , .(t, x, τ ) I−→

⎧ ∞ ⎪ ⎨ (i) Bij kh ∈ L ((0, T ) × Ω); . (ii) B(t)σ · τ = σ · B(t)τ ⎪ ⎩ ∀σ, τ ∈ Sd , a.e. t ∈ (0, T ), a.e. in Ω

(15)

where we denote by .B(t)τ = B(t, ·, τ ) which is defined on .Ω for every .t ∈ (0, T ), τ ∈ Sd . We suppose the body forces and surface tractions satisfy

.

f0 ∈ L2 (0, T ; H ),

.

f2 ∈ L2 (0, T ; L2 (Γ2 )d )

(16)

On the contact surface, the following frictional contact function .ψc : (0, T ) × V −→ R,  .ψc (t, w) := ϕc (t, w) da, ∀(t, w) ∈ (0, T ) × V , Γ3

verifies ⎧ (i) t ∈ (0, T ) I−→ ψc (t, w) is Lebesgue measurable ∀w ∈ V ; ⎪ ⎪ ⎨ . (ii) |ψc (t, w)| ≤ c(t) + d ‖w‖V , ∀w ∈ V , a.e. t ∈ (0, T ); ⎪ ⎪ ⎩ (iii) ψc (t, ·) is convex on V a.e. t ∈ (0, T ),

(17)

where .d > 0 is some constante and .c ∈ L2 (0, T ; R+ ). The thermal expansion tensor .Ce : (0, T ) × Ω × R −→ Sd verifies ⎧ (i) Ce (·, ·, ϑ) is measurable on (0, T ) × Ω, ∀ϑ ∈ R; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ii) there exists Le > 0 such that ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |Ce (t, x, ϑ1 ) − Ce (t, x, ϑ2 )| ≤ Le |ϑ1 − ϑ2 | ⎪ ⎨ ∀ϑ1 , ϑ2 ∈ R, a.e. (t, x) ∈ (0, T ) × Ω; . ⎪ ⎪ ⎪ ⎪ (iii) there exists c0Ce ∈ L∞ ((0, T ) × Ω; R+ ), c1Ce ≥ 0 such that ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |Ce (t, x, ϑ)| ≤ c0Ce (t, x) + c1Ce |ϑ|, ⎪ ⎪ ⎪ ⎩ ∀ϑ ∈ R, for a.e. (t, x) ∈ (0, T ) × Ω.

(18)

Here we recall the notation .Ce (t, ϑ) = Ce (t, ·, ϑ) some function defined on .Ω, for all .t ∈ (0, T ) and .ϑ ∈ R. The nonlinear function .Kc : (0, T ) × Ω × Rd −→ R satisfies:

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

⎧ ⎪ (i) Kc (·, ·, ξ ) is measurable on (0, T ) × Ω, ∀ξ ∈ Rd ; ⎪ ⎪ ⎪ ⎪ ⎪ d ⎪ ⎪ (ii) Kc (t, x, ·) is continuous on R , a.e. (t, x) ∈ (0, T ) × Ω; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (iii) there exists c0Kc ∈ L2 ((0, T ) × Ω; R+ ), c1Kc ≥ 0, such that ⎪ ⎪ ⎪ ⎪ |Kc (t, x, ξ )| ≤ c0Kc (t, x) + c1Kc |ξ |, ⎪ ⎨ ∀ξ ∈ Rd , a.e. (t, x) ∈ (0, T ) × Ω; . ⎪ ⎪ (iv) there exists mKc > 0 such that ⎪ ⎪ ⎪ ⎪ (Kc (t, x, ξ1 ) − Kc (t, x, ξ2 )) · (ξ1 − ξ2 ) ≥ mKc |ξ1 − ξ2 |2 , ⎪ ⎪ ⎪ ⎪ ⎪ ∀ξ1 , ξ2 ∈ Rd , a.e. (t, x) ∈ (0, T ) × Ω ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (v) there exists nKc > 0 such that Kc (t, x, ξ ) · ξ ≥ nKc |ξ |2 , ⎪ ⎪ ⎩ ∀ξ ∈ Rd , a.e. (t, x) ∈ (0, T ) × Ω.

25

(19)

The viscosity-deformation heat function .De : (0, T ) × Ω × Rd −→ R satisfies: ⎧ (i) De (·, ·, v) is measurable on (0, T ) × Ω, ∀v ∈ Rd ; ⎪ ⎪ ⎪ ⎪ ⎨ (ii) there exists LDe > 0 such that . ⎪ ⎪ |De (t, x, v1 ) − De (t, x, v2 )| ≤ LDe |v1 − v2 |, ⎪ ⎪ ⎩ ∀v1 , v2 ∈ Rd , a.e. (t, x) ∈ (0, T ) × Ω.

(20)

We assume for the heat sources density, that q ∈ L2 (0, T ; L2 (Ω))

.

(21)

The nonlinear function .ϕthermal : (0, T ) × Γc × R −→ R verifies: ⎧ (i) ϕthermal (·, ·, r) is measurable on (0, T ) × Γc , ∀r ∈ R; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (ii) ϕthermal (t, x, ·) is convex derivable on R for a.e. (t, x) ∈ (0, T ) × Γc ; ⎪ ⎨ ϕ ϕ 2 + . ⎪ (iii) there exists c0 ∈ L ((0, T ) × Γc ; R ), c1 ≥ 0, such that ⎪ ⎪ ϕ ϕ ' ⎪ ⎪ |ϕthermal (t, x, r)| ≤ c0 (t, x) + c1 |r|, ⎪ ⎪ ⎩ ∀r ∈ R, a.e. (t, x) ∈ (0, T ) × Γc . (22) We notice that these assumptions are verified for the example (12). Finally we assume that the initial data satisfy the conditions u0 ∈ H,

.

v0 ∈ V ,

θ0 ∈ E.

(23)

To continue, using Green’s formula, we obtain the variational formulation of the mechanical problem Q in abstract form as follows. Problem QV Find .u : [0, T ] → V , .θ : [0, T ] → E satisfying a.e. .t ∈ (0, T ):

26

O. Chau et al.

⎧ 〈u(t) ¨ + A(t) u(t) ˙ + B(t) u(t) + C(t) θ (t), w − u(t)〉 ˙ ⎪ V ' ×V ⎪ ⎪ ⎨  t . +( B(t − s) ε(u(s)) ds, ε(w) − ε(u(t))) ˙ ˙ τ (t)) H + ψc (t, wτ ) − ψc (t, u ⎪ ⎪ 0 ⎪ ⎩ ≥ 〈f (t), w − u(t)〉 ˙ ∀w ∈ V . V ' ×V (24) 〈θ˙ (t), η〉E ' ×E + 〈K(t) θ (t), η〉E ' ×E + 〈ψthermal (t) θ (t), η〉E ' ×E . (25) = 〈R(t)u(t), ˙ η〉E ' ×E + 〈Q(t), η〉E ' ×E , ∀ η ∈ E. u(0) = u0 ,

u(0) ˙ = v0 ,

.

θ (0) = θ0 .

(26)

Here, the operators and functions .A(t), B(t) : V −→ V ' , .C(t) : E −→ V ' , ' ' ' ' .f : [0, T ] −→ V , .K(t) : E −→ E , .ψthermal (t) : E −→ E , .R(t) : V −→ E ' and .Q : [0, T ] −→ E are defined by .∀v ∈ V , .∀w ∈ V , .∀ζ ∈ E, .∀η ∈ E, a.e. .t ∈ (0, T ): 〈A(t) v, w〉V ' ×V = (A (t) ε(v), ε(w))H ;

(27)

〈B(t) v, w〉V ' ×V = (G (t) ε(v), ε(w))H ;

(28)

〈C(t)ζ, w〉V ' ×V = (Ce (t, ζ ), ε(w))H ;

(29)

〈f (t), w〉V ' ×V = (f0 (t), w)H + (f2 (t), w)(L2 (Γ2 ))d ;

(30)

.

.

.

.

 〈K(t) ζ, η〉

E ' ×E

.

 〈ψthermal (t) ζ, η〉E ' ×E =

.

Γ3

=

Kc (t, ∇ζ ) · ∇η dx;

(31)

' ϕthermal (t, x, ζ (x)) η(x) da(x);

(32)

Ω

 〈R(t) v, η〉E ' ×E =

De (t, v) η dx;

.

(33)

Ω

 〈Q(t), η〉

.

E ' ×E

=

q(t) η dx.

(34)

Ω

Our main existence and uniqueness result is stated as follows, that we prove in the next Section. Theorem 1 Assume that (13)–(23) hold, then there exists an unique solution .{u, θ } to the problem QV with the regularity: .

u ∈ W 1,2 (0, T ; V ) ∩ W 2,2 (0, T ; V ' ) ∩ C 1 (0, T ; H ) θ ∈ L2 (0, T ; E) ∩ W 1,2 (0, T ; E ' ) ∩ C(0, T ; F ).

(35)

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

27

Before proving the main theorem, we present here some examples with subdifferential friction laws of the form (6), see e.g. the monograph [6] or the Habilitation thesis [1] p. 117. Example 1 Contact with Tresca’s friction law. This contact condition can be found in [4, 7]. It is in the form of the following boundary condition: ⎧ ⎨ |σ τ | ≤ g, . |σ | < g =⇒ u˙ τ = 0, ⎩ τ |σ τ | = g =⇒ u˙ τ = −λσ τ , λ ≥ 0

on

Γ3 × (0, T ).

(36)

Here .g ∈ L∞ (Γ3 ; R+ ) represents the friction bound, i.e., the magnitude of the limiting friction traction at which slip begins. Then we define here ϕc (x, y) = g(x) |y|, ∀(x, y) ∈ Γ3 × Rd .

.

Example 2 Contact with viscoelastic friction condition. We consider the problems with the boundary conditions σ τ = −μ|u˙ τ |p−1 u˙ τ

.

on

Γ3 × (0, T ),

(37)

where .μ ∈ L∞ (Γ3 ; R+ ) is the coefficient of friction and .0 < p ≤ 1. Here, the tangential shear is proportional to the power p of the tangential speed, which is the case when the contact surface is lubricated with a thin layer of non-Newtonian fluid. We define here ϕc (x, y) =

.

μ(x) p+1 |y| , ∀(x, y) ∈ Γ3 × Rd . p+1

3 Proof of Theorem 1 The idea is to bring the second order inequality to a first order inequality, using monotone operator, convexity and fixed point arguments, and will be carried out in several steps. Let us introduce the velocity variable v = u. ˙

.

The system in Problem QV is then written for a.e. .t ∈ (0, T ):

28

O. Chau et al.

⎧

t u(t) = u0 + 0 v(s) ds; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 〈v(t) ˙ + A(t) v(t) + B(t) u(t) + C(t) θ (t), w − v(t)〉V ' ×V ⎪ ⎪ ⎪  t ⎪ ⎪ ⎪ ⎪ ⎪ +( B(t − s) ε(u(s)) ds, ε(w) − ε(v(t)))H + ψc (t, w) − ψc (t, v(t)) ⎨ .

0

≥ 〈f (t), w − v(t)〉V ' ×V ∀w ∈ V ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 〈θ˙ (t), η〉E ' ×E + 〈K(t) θ (t), η〉E ' ×E + 〈ψthermal (t) θ (t), η〉E ' ×E ⎪ ⎪ ⎪ ⎪ = 〈R(t)u(t), ˙ η〉E ' ×E + 〈Q(t), η〉E ' ×E , ∀ η ∈ E; ⎪ ⎪ ⎪ ⎪ ⎩ v(0) = v0 , θ (0) = θ0 ,

with the regularity .

v ∈ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ' ) ∩ C(0, T ; H ) θ ∈ L2 (0, T ; E) ∩ W 1,2 (0, T ; E ' ) ∩ C(0, T ; F ).

To continue, we assume in the sequel that the conditions (13)–(17) of the Theorem 1 are satisfied. Let define W := L2 (0, T ; H ).

.

We begin by Lemma 1 For all .η ∈ W , there exists an unique vη ∈ L2 (0, T ; V ) ∩ W 1,2 (0, T ; V ' ) ∩ C(0, T ; H )

.

satisfying ⎧ 〈v˙η (t) + A(t) vη (t), w − vη (t)〉V ' ×V + (η(t), ε(w) − ε(vη (t)))H ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ +ψc (t, w) − ψc (t, vη (t)) ≥ 〈f (t), w − vη (t)〉V ' ×V , .

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∀ w ∈ V,

a.e. t ∈ (0, T );

(38)

vη (0) = v0 .

Moreover, .∃c > 0 such that .∀η1 , η2 ∈ W :  ‖vη2 (t)−vη1 (t)‖2H +

t

.

0

 ‖vη1 −vη2 ‖2V ≤ c

0

t

‖η1 −η2 ‖2H ,

∀t ∈ [0, T ].

(39)

Proof Let .η ∈ W . Using [9] II/B p. 893, we deduce the existence and uniqueness of .vη .

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

29

Now let .η1 , η2 ∈ W . In (38) we take .(η = η1 , w = vη2 (t)), then .(η = η2 , w = vη1 (t)). Adding the two inequalities, we deduce that for a.e. .t ∈ (0; T ): 〈v˙η2 (t) − v˙η1 (t), vη2 (t) − vη1 (t)〉V ' ×V + 〈A(t) vη2 (t) .

−A(t) vη1 (t), vη2 (t) − vη1 (t)〉V ' ×V ≤ −(η2 (t) − η1 (t), ε(vη2 (t)) − ε(vη1 (t)))H .

Then integrating over .(0, t), from (13)(iii) and from the initial condition on the velocity, we obtain:

t ∀t ∈ [0, T ], ‖vη2 (t) − vη1 (t)‖2H + mA 0 ‖vη2 (s) − vη1 (s)‖2V ds  t . ≤− (η2 (s) − η1 (s), ε(vη2 (s)) − ε(vη1 (s)))H ds

0 t +mA 0 ‖vη2 (s) − vη1 (s)‖2H ds. We conclude that .∃c > 0 such that .∀η1 , η2 ∈ W , .∀t ∈ [0, T ]:

.

2 ‖vη2 (t)  − vη1 (t)‖H + t

≤c 0

t 0

‖vη1 (s) −vη2 (s)‖2V ds t

‖η1 (s) − η2 (s)‖2H ds + c

0

‖vη2 (s) − vη1 (s)‖2H ds.

(40)

Now let fix .τ ∈ [0, T ]. We have .∀t ∈ [0, τ ]:  ‖vη2 (t) − vη1 (t)‖2H ≤ c

.

0

τ

 ‖η1 (s) − η2 (s)‖2H + c

t 0

‖vη2 (s) − vη1 (s)‖2H ds.

Using then Gronwall’s inequality, we obtain .∀τ ∈ [0, T ]: 2 .‖vη2 (τ ) − vη1 (τ )‖H

  ≤ c 0

τ

 ‖η1 (s) − η2 (s)‖2H ecT .

Finally, integrating the last inequality and reporting the result in (40), we get (39). Here and below, we denote by .c > 0 a generic constant, which value may change from lines to lines. Lemma 2 For all .η ∈ W , there exists an unique θη ∈ L2 (0, T ; E) ∩ W 1,2 (0, T ; E ' ) ∩ C(0, T ; F )

.

satisfying

30

O. Chau et al.

⎧ ⎪ θ˙ (t) + K(t) θη (t) + ψthermal (t) θη (t) ⎪ ⎨ η = R(t)vη (t) + Q(t), ∀ η ∈ E, a.e. t ∈ (0, T ); . ⎪ ⎪ ⎩ θη (0) = θ0 .

(41)

Moreover, .∃c > 0 such that .∀η1 , η2 ∈ W :  ‖θη1 (t) − θη2 (t)‖2F ≤ c

t

.

0

‖vη1 − vη2 ‖2V ,

∀t ∈ [0, T ].

(42)

Proof The existence and uniqueness result verifying (41) follows from standard result on first order evolution equation (see e.g. [5]). Indeed we verify that from the expression of the operator R, we have vη ∈ L2 (0, T ; V ) =⇒ R vη ∈ L2 (0, T ; E ' ),

.

as .Q ∈ L2 (0, T ; E ' ) then .R vη + Q ∈ L2 (0, T ; E ' ). Using the assumptions (19) and (22), the operator K(t) + ψthermal (t) : E −→ E '

.

for a.e. .t ∈ (0, T ) is strongly monotone. Now for .η1 , η2 ∈ W , we have for a.e. .t ∈ (0; T ): 〈θ˙η1 (t) − θ˙η2 (t), θη1 (t) − θη2 (t)〉E ' ×E .

+〈K(t) θη1 (t) − K(t) θη2 (t), θη1 (t) − θη2 (t)〉E ' ×E ≤ 〈R(t) vη1 (t) − R(t) vη2 (t), θη1 (t) − θη2 (t)〉E ' ×E .

Then integrating the last property over .(0, t), using the strong monotonicity of .K(t) and the Lipschitz continuity of .R(t) : V −→ E ' , we deduce (42). Proof of Theorem 1 We have now all the ingredients to prove the Theorem 1. Consider the operator .Λ : W → W defined by for all .η ∈ W : 

t

Λ η (t) = G (ε(uη (t))) +

.

B(t − s) ε(uη (s)) ds + Ce (t, θη (t)),

∀t ∈ [0, T ],

0

where  uη (t) = u0 +

.

t

vη (s) ds, ∀t ∈ [0, T ];

0

uη ∈ W 1,2 (0, T ; V ) ∩ W 2,2 (0, T ; V ' ) ∩ C 1 (0, T ; H ).

A Class of Dynamic Unilateral Contact Problems with Sub-differential Friction Law

31

Then from (14), (15), and Lemma 2, we deduce that for all .η1 , η2 ∈ W , for all t ∈ [0, T ]:

.

 ‖Λ η1 (t) − Λ η2 (t)‖2H ≤ c ‖θη1 (t) − θη2 (t)‖2F + c  t . ≤c ‖vη1 (s) − vη2 (s)‖2V ds.

t 0

‖vη1 (s) − vη2 (s)‖2V ds

0

(43) Now using (43), after some algebraic manipulations, we have for any .β > 0: 

T

.

0

e−βτ ‖Λ η1 (τ ) − Λ η2 (τ )‖2H ≤

c β

 0

T

e−βτ ‖η1 (τ ) − η2 (τ )‖2H dτ.

We conclude from the last inequality by contracting principle that the operator .Λ has a unique fixed point .η∗ ∈ W . We verify then that the functions  u(t) := u0 +

.

t

vη∗ , ∀t ∈ [0, T ],

θ := θη∗

0

are solutions to problem QV with the regularity (35), the uniqueness follows from the uniqueness in Lemma 1 and Lemma 2.

References 1. O. Chau, Quelques problèmes d’évolution en mécanique de contact et en biochimie, Habilitation thesis, Saint Denis La Réunion, University Press (2010) 2. O. Chau, A. Heibig, A. Petrov, Solvability for a class of unilateral contact problems with friction and damage, in Mathematical Analysis, Differential Equations and Applications, ed. by T.M. Rassias (WSPC Singapore, Singapore, to appear) 3. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Contributions to Nonlinear Functional Analysis (North Holland, Amsterdam, 1978) 4. G. Duvaut, J.L. Lions, Les Inéquations en Mécanique et en Physique (Dunod, Paris, 1972) 5. J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Contributions to Nonlinear Functional Analysis (Gauthier-Villars, Dunod, 1969) 6. P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Contributions to Nonlinear Functional Analysis (Birkhäuser, Basel, 1985) 7. P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering. Contributions to Nonlinear Functional Analysis (Springer, Berlin, 1993) 8. A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. e Appl. 18, 95–139 (1959) 9. E. Zeidler, Nonlinear Functional Analysis and its Applications. Contributions to Nonlinear Functional Analysis (Springer, Berlin, 1997)

Square-Free Values of [nc tanθ (log n)] .

S. I. Dimitrov

2020 Mathematics Subject Classification 11L07, 11N25

1 Notations Let x be a sufficiently large positive number. By .ε we denote an arbitrary small positive constant, not the same in all appearances. We denote by .μ(n) the Möbius function. As usual .[t] and .{t} denote the integer part, respectively, the fractional part of t. Moreover .e(y) = e2π iy . Instead of .m ≡ n (mod k) we write for simplicity 1 3 .m ≡ n (k). Throughout this paper we suppose that .1 < c < 2 and .γ = c . Assume that .θ > 1 is a fixed. Denote z=x

.

2c−1 4

;.

ψ(t) = {t} − 1/2 ; .  log x  Δ1 = eπ π +arctan 1 ; .   π logπ x +arctan 2 ;. Δ2 = e    Sc (x) = μ2 [nc tanθ (log n)] .

(1) (2) (3) (4) (5)

Δ1 ≤n 1 is a fixed, then the lower bound  . Δ1 ≤n 1 is a fixed. Then for the sum (5) the lower Sc (x) ⪢ x

(7)

.

holds.

3 Preliminary Lemmas Lemma 1 For every H ≥ 1, we have 

ψ(t) =

.

a(h)e(ht) + O



b(h)e(ht) ,

|h|≤H

1≤|h|≤H

where a(h) ⪡

.

1 , |h|

b(h) ⪡

1 . H

(8)

Here ψ(t) is denoted by (2). Proof See [7]. Lemma 2 Let q ≥ 0 be an integer. Suppose that f (t) has q + 2 continuous derivatives on I , and that I ⊆ (N, 2N]. Assume also that there is some constant F such that |f (r) (t)| ≍ F N −r

.

for r = 1, . . . , q + 2. Let Q = 2q . Then   q+2   1 . e(f (n)) ⪡ F 4Q−2 N 1− 4Q−2 + F −1 N .  n∈I

The implied constant depends only upon the implied constants in (9). Proof See ([4], Theorem 2.9). The next lemma is well-known.

(9)

36

S. I. Dimitrov

Lemma 3 Let the function f (x) be continuously differentiable, nonnegative and monotonic in the interval [a, b]. Then     b       . f (n) − f (x) dx  ≤ max f (a), f (b) .   a s0 , then we should have, on one hand, .ϕ(s1 ) = R0 by definition of .s1 and, on the other hand, ' .ϕ(s1 ) = lim s→s + ϕ(s) ≤ R < R0 , which is impossible). This establishes the first 1

74

L. Fresse and V. V. Motreanu

part of (17). The second part of (17) now follows from (15) and (18). This shows (17). We now claim that .s0 = −∞ (and consequently .r0 = −∞). Arguing by contradiction, assume that .s0 > −∞. By (17), .ϕ is increasing on .(s0 , r] so that L := lim ϕ(s) exists and belongs to [0, R ' ].

.

s→s0+

This fact guarantees that the solution .ϕ can be extended on the left of .s0 , hence we must have .r0 < s0 . Then, by the minimality of .s0 , we must have .L = ϕ(s0 ) = 0. This yields  .

0

R'



1 ds = kα(s) − 𝓁s

r

s0

ϕ ' (s) ds = r − s0 < +∞ kα(ϕ(s)) − 𝓁ϕ(s)

(by (16)). However, using (14), we can see that (4) implies that  .

0

R'



1 ds ≥ kα(s) − 𝓁s

R'

0

1 ds = +∞, 2kα(s)

a contradiction. We conclude that .s0 = r0 = −∞. Thus (17) holds on the whole interval .(−∞, r]. Finally, we define .σ ∈ C 2 ([0, r], R) by  σ (t) =

t

ϕ(s) ds.

.

0

This definition combined with (16) and (17) ensures that conditions (10), (11), and (12) are fulfilled. The final assertion of the lemma can be justified as follows. Relation (12) guarantees that .σ and .σ ' are increasing on .[0, r], and moreover 

t

σ (t) =

.

σ ' (s) ds ≤ tσ ' (t) ≤ rσ ' (t) ≤ rR,

0

whence (13). The proof is complete.

⨆ ⨅

Proof of Lemma 1 By assumption (3), we find constants .μ > 0 and .δ ∈ (0, δ0 ) such that .

max{β(x, s), χ (x, s)} < μ for all x ∈ Ω and all s ∈ (0, δ). φ(x, s)α(s)

(19)

A Strong Maximum Principle for General Nonlinear Operators

75

Since .u(z) = 0 and u is continuous at z, there exists r with .0 < r < min{ ρ2 , 2δ , 1} such that u(x) < δ whenever |x − z| ≤ 4r.

.

−z , we have that Letting .z0' = z + 2r |zz00 −z|

z ∈ ∂B2r (z0' ) and B2r (z0' ) \ {z} ⊂ Bρ (z0 ) ⊂ {x ∈ Ω : u(x) > 0}

.

(20)

(see (9)). Up to applying a translation, we may assume that .z0' = 0. Let Ω0 = {x ∈ Ω : r < |x| < 2r}.

.

Thus we have sup u(x) ≤ δ < δ0 and sup |∇u(x)| ≤ M,

.

x∈Ω0

(21)

x∈Ω0

for some .M > 0. We apply Lemma 2 to r and the constant 𝓁 :=

.

 Nc 1  c1 − − Nc − Mc r c1 2r

for .c1 , c from Assumption 1(d). We choose .k ∈ (k0 , +∞) (for the constant .k0 provided by Lemma 2) such that Nc + Mc + c0
0 in (8) and .μ in (19)). Next we choose .R ∈ (0, R0 ) (for the constant .R0 provided by Lemma 2) such that 1 .

r

+

 Rc 1 kc1 + Nc + Mc + c0 < +N 2 r 2μ r

(22)

and R < δ and R ≤ min u

.

∂Br (0)

(23)

(recall (20)). Lemma 2 now yields a function .σ : [0, r] → [0, R) of class .C 2 satisfying (10)–(12). Then let .v : Ω0 → [0, R) be given by v(x) = σ (2r − |x|) for all x ∈ Ω0

.

76

L. Fresse and V. V. Motreanu

so that .v ∈ C 2 (Ω0 ). Note that .v ≡ 0 on .∂B2r (0) and .v < R on .∂Br (0), whence v ≤ u on ∂Ω0

(24)

.

(see (23)). Our aim is to compare u with v in .Ω0 . The first step is the following claim. Claim 1 For some constant .γ > 0, the following inequality holds in .Ω0 , in the sense of distributions: −div a(x, u, ∇v(x)) + div a(x, u, ∇u(x))     ≤ γ β(x, u(x)) − β(x, v(x)) + c0 χ (x, |∇u(x)|) − χ (x, |∇v(x)|) .

.

Proof of Claim 1 Fix .x = (xi )N i=1 ∈ Ω0 and set .t = 2r − |x| ∈ (0, r). We see that ∂v ' (t) xi (.i ∈ {1, . . . , N }), whence (x) = −σ . ∂xi |x| |∇v(x)| = σ ' (t) ∈ (0, δ)

(25)

.

(by (12) and (23)), and .

 ∂ 2v σ ' (t) 1  σ ' (t) '' + σ δi,j , (t) x x − (x) = i j |x| ∂xi ∂xj |x|2 |x|

where .δi,j stands for the Kronecker’s symbol (.i, j ∈ {1, . . . , N}). Moreover, we have v(x) = σ (t) ≤ rσ ' (t) < σ ' (t)

(26)

.

(see Lemma 2, recalling that .r < 1). Denoting by .ai the i-th component of the map a, we can write div a(x, u, ∇v(x)) =

.

N   ∂  ai (x, u, ∇v) ∂xi i=1

=

N  ∂ai i=1

+

∂xi

(x, u, ∇v) +

N  ∂ai i=1

∂s

(x, u, ∇v)

N  ∂ai ∂ 2v (x). (x, u, ∇v) ∂xi ∂xj ∂ξj

i,j =1

∂u (x) ∂xi

A Strong Maximum Principle for General Nonlinear Operators

77

N We estimate each of the above three terms. Let .ei = (δi,j )N j =1 ∈ R . By (7) combined with (20), (21) and (25) we have

.

N  ∂ai i=1

∂xi

(x, u, ∇v) =

N 

ax' (x, u, ∇v)ei · ei

i=1

  ≥ −Nc φ(x, |∇v(x)|)|∇v(x)| + β(x, u(x)) = −Ncφ(x, σ ' (t))σ ' (t) − Ncβ(x, u(x))

and

.

N  ∂ai i=1

∂s

(x, u, ∇v)

N  ∂u ∂u (x) = as' (x, u, ∇v) · ei (x) ∂xi ∂xi i=1

= as' (x, u, ∇v) · ∇u(x)   ≥ −c φ(x, |∇v(x)|)|∇v(x)| + β(x, u(x)) |∇u(x)| ≥ −Mcφ(x, σ ' (t))σ ' (t) − Mcβ(x, u(x)). Finally, using (5) (since .r ≤ |x| ≤ 2r < δ0 ), (6), in conjunction with (20), (21), and (25), we deduce that N  ∂ai ∂ 2v (x) (x, u, ∇v) ∂xi ∂xj ∂ξj

.

i,j =1

=

N 

aξ' (x, u, ∇v)ej · ei

i,j =1

= aξ' (x, u, ∇v)x · x

 1  σ ' (t)  σ ' (t)  '' + σ δi,j (t) x x − i j |x| |x|2 |x|

  σ ' (t) 1  σ ' (t) '' + σ (t) − aξ' (x, u, ∇v)ei · ei 2 |x| |x| |x| N

i=1

  1  σ ' (t) '' ≥ + σ (t) c1 φ(x, |∇v(x)|) |x|2 − cβ(x, u(x)) 2 |x| |x|   σ ' (t) Nc φ(x, |∇v(x)|) + β(x, u(x)) |x|     σ ' (t) σ ' (t) Nc + σ '' (t) − = φ(x, σ ' (t)) c1 |x| |x|  c  σ ' (t)  σ ' (t)  '' −β(x, u(x)) (t) + + σ Nc |x| |x|2 |x| −

78

L. Fresse and V. V. Motreanu

 Nc  ' σ (t) + c1 σ '' (t) 2r r  c  Nc  ' c −β(x, u(x)) + σ (t) + 2 σ '' (t) . 3 r r r

≥ φ(x, σ ' (t))

 c

1

−

Altogether, inserting these three estimates in the preceding formula and using (10) and (12), we get  c   Nc 1 div a(x, u, ∇v(x)) ≥ φ(x, σ ' (t)) c1 σ '' (t) + − − Nc − Mc σ ' (t) 2r r c  c Nc  ' σ (t) + Nc + Mc −β(x, u(x)) 2 σ '' (t) + 3 + r r r

.

≥ kc1 φ(x, σ ' (t))α(σ ' (t)) c c  Nc  −β(x, u(x)) 2 R + 3 + R + Nc + Mc r r r for all .x ∈ Ω0 . On the other hand, using (19) (recall (25)) and the fact that the function .β(x, ·) is nondecreasing (see Assumption 1(b)) combined with (26), we have φ(x, σ ' (t))α(σ ' (t)) ≥

.

1 1 β(x, σ ' (t)) ≥ β(x, v(x)) for all x ∈ Ω0 . μ μ

In addition, using (19) and (25), we get also φ(x, σ ' (t))α(σ ' (t)) ≥

.

1 1 χ (x, σ ' (t)) = χ (x, |∇v(x)|) for all x ∈ Ω0 . μ μ

It follows that div a(x, u, ∇v(x)) ≥

.

kc1 kc1 β(x, v(x)) + χ (x, |∇v(x)|) 2μ 2μ  Rc   1 1 + 2 +N + Nc + Mc −β(x, u(x)) r r r

for all .x ∈ Ω0 . The choice of k and R in (22) allows us to find a constant .γ such that 1 .

r

+

 Rc kc1 1 +N + Nc + Mc + c0 < γ < . 2 r 2μ r

A Strong Maximum Principle for General Nonlinear Operators

79

1 Also note that .c0 < kc 2μ . Using again that the functions .β and .χ are nonnegative as well as hypothesis (8) in conjunction with (20) and (21), we derive that

div a(x, u, ∇v(x)) ≥ γβ(x, v(x)) + (c0 − γ )β(x, u(x))

.

+c0 χ (x, |∇v(x)|) ≥ γβ(x, v(x)) − γβ(x, u(x)) +c0 χ (x, |∇v(x)|) − c0 χ (x, |∇u(x)|) +div a(x, u, ∇u(x)) in .Ω0 , where the last inequality holds in distributional sense. This establishes Claim 1. ⨆ ⨅ Since .u, v ∈ C 1 (Ω0 ) ⊂ W 1,1 (Ω0 ), we get that w := max{v − u, 0} ∈ W 1,1 (Ω0 ) ∩ C(Ω0 ).

.

Since .v ≤ u on .∂Ω0 (see (24)), we have that .w|∂Ω0 = 0, which implies that .w ∈ W01,1 (Ω0 ), .w ≥ 0. In fact, we claim the following: Claim 2 .w ≡ 0, that is, .v ≤ u in .Ω0 . Proof of Claim 2 Arguing by contradiction, assume that .w /≡ 0. Then, since .w ∈ C(Ω0 ), .w ≥ 0, we have m0 := max w > 0.

(27)

.

Ω0

For every .m ∈ (0, m0 ), we let Ωm = {x ∈ Ω0 : m < w(x) ≤ m0 },

.

which is a nonempty open subset of .Ω0 . Subclaim 1 We can find .m1 ∈ (0, m0 ) and a compact subset .K = K1 ×K2 ×K3 ⊂ Ω0 × (0, δ0 ) × (Bδ0 (0) \ {0}) such that, for every .m ∈ (m1 , m0 ), we have Ωm ⊂ K1 ,

.

u(x), v(x) ∈ K2 ,

∇u(x), ∇v(x) ∈ K3

for all x ∈ Ωm .

Proof of Subclaim 1 First we note that, since .w|∂Ω0 = 0, .Ωm is a compact subset of .Ω0 for every .m ∈ (0, m0 ). The equality .w = v − u holds in .Ωm so that w is there of class .C 1 . Moreover, .Ωm contains all the points .x ∈ Ω0 where the maximum .m0 of w is attained, and we have .∇w(x) = 0 for every such point x. This easily yields .

lim max |∇v(x) − ∇u(x)| = lim max |∇w(x)| = 0.

m→m0 Ωm

m→m0 Ωm

(28)

80

L. Fresse and V. V. Motreanu

Fix .m2 ∈ (0, m0 ). By (25) there is an interval .[δ1 , δ2 ] ⊂ (0, δ0 ) such that |∇v(x)| ∈ [δ1 , δ2 ] for all x ∈ Ωm2 .

.

Due to (28), by choosing .m1 ∈ (m2 , m0 ) close enough to .m0 , we will have 2 |∇u(x)|, |∇v(x)| ∈ [ δ21 , δ0 +δ 2 ]

.

for all x ∈ Ωm1 .

2 We set .K3 = {ξ ∈ RN : δ21 ≤ |ξ | ≤ δ0 +δ 2 }. Since .0 < u(x) < v(x) < δ0 for all .x ∈ Ωm1 (see (20), (25), (26)), the set .K2 := u(Ωm1 ) ∪ v(Ωm1 ) is a compact subset of .(0, δ0 ). Finally we set .K1 = Ωm1 . The triple .(K1 , K2 , K3 ) so obtained satisfies the properties stated in the subclaim. ⨆ ⨅

Considering the compact set K of Subclaim 1, let .κ = κ(K) be the constant provided by Assumption 1(a), so that (a(x, s, ξ ) − a(x, s, η)) · (ξ − η) > κ|ξ − η|2

.

(29)

for all .x ∈ K1 , .s ∈ K2 , .ξ, η ∈ K3 , .ξ /= η. Let .L = L(K1 × {|ξ | : ξ ∈ K3 }) be the constant provided by Assumption 1(b) so that |χ (x, s) − χ (x, t)| ≤ L|s − t| for all x ∈ K1 , s, t ∈ {|ξ | : ξ ∈ K3 }.

.

(30)

0 ⊂ Ω in In what follows, we fix .m ∈ (m1 , m0 ) and a connected component .Ωm m which the maximal value .m0 of w is attained.

Subclaim 2 The following inequality holds:  . 0 Ωm

(a(x, u, ∇v) − a(x, u, ∇u)) · (∇v − ∇u) dx



≤γ

0 Ωm

(β(x, u) − β(x, v))(w − m) dx



+c0

0 Ωm

(χ (x, |∇u|) − χ (x, |∇v|))(w − m) dx.

Proof of Subclaim 2 Since .Ωm ⊂ Ω0 , we must have .w ≡ m on .∂Ωm , hence 0 (w − m)|Ωm0 ∈ W01,1 (Ωm ).

.

0 ), .w ≥ 0, such that This implies that there is a sequence .(wn ) ⊂ Cc∞ (Ωm n 0 wn → w − m in W01,1 (Ωm ) as n → ∞

.

(31)

A Strong Maximum Principle for General Nonlinear Operators

81

(see [1, p. 259–261]). By Claim 1, we have  . 0 Ωm

(a(x, u, ∇v) − a(x, u, ∇u)) · ∇wn dx



≤ c0

0 Ωm

(χ (x, |∇u|) − χ (x, |∇v|)wn dx



+γ

0 Ωm

(β(x, u) − β(x, v))wn dx.

(32)

0 ⊂ Ω , .u, v ∈ C 1 (Ω ), and a, .β, .χ are continuous, we have Since .Ωm 0 0 0 , RN ) ⊂ L∞ (Ω 0 , RN ) a(x, u, ∇v) − a(x, u, ∇u) ∈ C(Ωm m

.

and 0 ) ⊂ L∞ (Ω 0 ). β(x, u) − β(x, v), χ (x, |∇u|) − χ (x, |∇v|) ∈ C(Ωm m

.

In view of (31), we can pass to the limit in (32), so that we obtain the formula stated in the subclaim. ⨆ ⨅ 0 (since .∂Ω 0 ⊂ ∂Ω and .w| Note that w is not constant on .Ωm m ∂Ωm = m < m0 = m 0 0. maxΩm0 w), hence .{x ∈ Ωm : ∇u(x) = / ∇v(x)} is a nonempty open subset of .Ωm By (29), the left-hand side of the formula of Subclaim 2 is positive. In the case where .χ ≡ 0, the assumption that .β(x, ·) is nondecreasing for all x implies that the right-hand side of the formula of Subclaim 2 is nonpositive and we reach a contradiction. It remains to consider the case where .χ /≡ 0, in which case we have .κ > 0 due to Assumption 1 (c). Invoking (29), (30), and the fact that .β(x, ·) is nondecreasing for all x, we derive from Subclaim 2 that   .κ |∇v − ∇u|2 dx ≤ c0 L |∇v − ∇u|(w − m) dx. 0 Ωm

0 Ωm

0 : w(y) = m }. Letting .Ω ˆ m = {x ∈ In view of (27), we have .∇v = ∇u on .{y ∈ Ωm 0 0 0 Ωm : m < w(x) < m0 } and .Ωˆ m = Ωm ∩ Ωˆ m (which is a nonempty open subset of 0 .Ωm ), we get

 ‖∇w‖2L2 (Ω 0 ) =

.

m

≤

0 Ωˆ m

c0 L κ

|∇v − ∇u|2 dx  0 Ωˆ m

|∇v − ∇u|(w − m) dx

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L. Fresse and V. V. Motreanu

c0 L ‖w − m‖L2 (Ωˆ 0 ) ‖∇w‖L2 (Ωˆ 0 ) m m κ θ−2 c0 L ≤ |Ωˆ m | 2θ ‖w − m‖Lθ (Ωm0 ) ‖∇w‖L2 (Ωm0 ) , κ ≤

(33)

where we have applied Hölder’s inequality, for some .θ ∈ (2, 2∗ ) with the Sobolev ∗ ˆ critical exponent .2∗ = N2N −2 if .N > 2 and .2 = +∞ if .N ≤ 2; hereafter, .|Ωm | N stands for the Lebesgue measure in .R . From the Sobolev inequality, there is a constant .S > 0 such that ‖h‖Lθ (Ω0 ) ≤ S‖∇h‖L2 (Ω0 )

.

for all h ∈ W01,2 (Ω0 ).

Applying this with .h = (w − m)1Ωm0 ∈ W01,2 (Ω0 ), where .1Ωm0 denotes the 0 of .Ω (see, e.g., [4, Proposition 1.61]), characteristic function of the subdomain .Ωm 0 we have ‖w − m‖Lθ (Ωm0 ) = ‖(w − m)1Ωm0 ‖Lθ (Ω0 )

.

≤ S‖∇((w − m)1Ωm0 )‖L2 (Ω0 ) = S‖∇w‖L2 (Ωm0 ) . From (33) we obtain ‖∇w‖2L2 (Ω 0 ) ≤

.

m

θ−2 c0 LS |Ωˆ m | 2θ ‖∇w‖2L2 (Ω 0 ) , m κ

hence 1≤

.

θ−2 c0 LS |Ωˆ m | 2θ κ

0 ). However, (since we have .‖∇w‖L2 (Ωm0 ) /= 0 because w is not constant on .Ωm ˆ m | = 0, hence we reach a contradiction. This shows Claim 2. .limm→m0 |Ω ⨆ ⨅

Now that we have Claim 2, the proof of the lemma can be concluded as follows. 2r Since .z = z − z0' = |z−z (z − z0 ), it remains to show that .∇u(z) · z < 0. Since 0| .u(z) = v(z) = 0 and .v ≤ u in .Ω0 (by Claim 2), we obtain ∇u(z) · z = − lim

.

t→0+

≤ − lim

t→0+

u((1 − t)z) t v((1 − t)z) d = − σ (2r − (1 − t)|z|)(0) t dt

= −|z|σ ' (0) < 0 (see (12)). The proof of Lemma 1 is complete.

⨆ ⨅

A Strong Maximum Principle for General Nonlinear Operators

83

Proof of Theorem 1 (a) Arguing by contradiction, assume that .Ω ' := {x ∈ Ω : u(x) > 0} /= Ω. Then, due to the continuity of u (and the connexity of .Ω), there is .z' ∈ Ω ∩ ∂Ω ' so that .u(z' ) = 0. Let .r0 > 0 be such that .B2r0 (z' ) ⊂ Ω. We can find .z0 ∈ Br0 (z' ) such that .u(z0 ) > 0. Note that the map m : (0, r0 ) → R, ρ I→ min u

.

Bρ (z0 )

is nonincreasing and continuous. Then, since .m(|z' − z0 |) = 0, there is .ρ ∈ (0, r0 ) minimal such that .m(ρ) = 0. This implies that .Bρ (z0 ) ⊂ Ω ' and there is ' .z ∈ ∂Bρ (z0 ) with .u(z) = 0. Note that .z ∈ Ω because .Bρ (z0 ) ⊂ B2r0 (z ) ⊂ Ω. Thus we are in position to apply Lemma 1, which provides the inequality ∇u(z) · (z − z0 ) < 0.

.

This implies that, whenever .t > 0 is small enough, we have .z + t (z − z0 ) ∈ Ω and u(z + t (z − z0 )) < u(z) = 0,

.

which contradicts the assumption that .u ≥ 0 in .Ω. We have therefore shown part (a) of the theorem. (b) Using that .x0 ∈ ∂Ω satisfies the interior sphere condition, we find .z0 ∈ Ω and .ρ > 0 such that .Bρ (z0 ) ⊂ Ω and Bρ (z0 ) ∩ ∂Ω = {x0 }.

.

Due to part (a), we have that .u(x) > 0 for all .x ∈ Bρ (z0 ). Hence we can apply Lemma 1 with .z = x0 , and we get that .

∂u x0 − z0 < 0. (x0 ) = ∇u(x0 ) · |x0 − z0 | ∂n

This shows part (b). The proof of the theorem is complete. ⨆ ⨅

References 1. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. (American Mathematical Society, Providence, 2010) 2. S. Miyajima, D. Motreanu, M. Tanaka, Multiple existence results of solutions for the Neumann problems via super- and sub-solutions. J. Funct. Anal. 262(4), 1921–1953 (2012)

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3. M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations. Nonlinear Anal. 37(4), 431–448 (1999) 4. D. Motreanu, V.V. Motreanu, N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems (Springer, New York, 2014) 5. P. Pucci, J. Serrin, The Maximum Principle. Progress in Nonlinear Differential Equations and Their Applications, vol. 73 (Birkhäuser, Basel, 2007) 6. J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984) 7. Q. Zhang, A strong maximum principle for differential equations with nonstandard p(x)-growth conditions. J. Math. Anal. Appl. 312(1), 24–32 (2005)

On the Application of Ergodic Theory to an Alternating Series Expansion for Real Numbers Chryssoula Ganatsiou and Ilias K. Savvas

2010 AMS Mathematics Subject Classification 11K55, 37A25, 37A30, 37A50, 37A10, 60A10

1 Introduction It is known that Computational Mathematics involves areas of mathematical works that focus on the applications of mathematics to computing technologies. This means that all the aspects of computational mathematics involves mathematical research in mathematics as well as in areas of science where computing plays a central and essential role emphasizing in algorithms, numerical methods and symbolic computations [12, 13, 32, 43]. Although the basic concepts and mathematical methods of computational mathematics serve to study and describe problems of various branches including computer computation in applied mathematics (computer algebra, effective methods, complexity theory etc.) they can be used in other research subjects such as the computational number theory also known as algorithmic number theory which deals with the study of performing number theoretic computations through suitable algorithms [1, 4, 40, 41, 45]. In particular algorithms are one of the most important subjects studied in computational mathematics because it is one of the most common tool for artificial structures in number theory in order to find number theoretic expansions. It is worth mentioning the notable contribution of Adrien-Marie Legendre in number theory, where he conjectured the quadratic reciprocity law, subsequently proved by Gauss. He also did innovative work on the distribution of primes and on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Furthermore the Gauss-Legendre algorithm for the computation of the digits of number .π being rapidly convergent

C. Ganatsiou () · I. K. Savvas Department of Digital Systems, University of Thessaly, Gaiopolis, Larissa, Greece e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_6

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with only 25 iterations producing 45 million correct digits of .π is an important effort in the approximation of number .π. The method is based on the individual work of Carl Friedrich Gauss (1777–1855) and Adrien-Marie Legendre (1752–1883) combined with modern algorithms for multiplication and square roots [31, 37]. It is also known that the study of number theoretic expansions have a long history which has been advanced to many directions of investigation involving domains outside Number Theory such as Algebra, Measure Theory, Stochastic Processes, Probability etc. [2, 6, 7, 35, 41]. Mathematical analysis has mostly introduced ideas in connection with algebraic or analytic functions which as long as they were influenced by engineering interests of logarithmic or exponential matter lead to number expansions in terms of either series, products or continued fractions. To this direction there is a wide variety of concepts especially in the Metrical Number Theory, where the main tool for dealing with them is the stochastic algorithms, namely randomized algorithms. These algorithms according to Kolmogorov’s program and last study [44] will play an important role in mathematics’ evolution as they lead to the chaos approach. They further reinforce the idea of approaching the context of probability through the entropy of random sequences, produced by stochastic algorithms [16]. Of particular interest are the algorithms that lead to alternating number expansions for real numbers in terms of rationals. The study of series representations for real numbers in terms of rationals had been a research subject for many mathematicians. In 1770 J. H. Lambert introduced two positive series expansions for real numbers in terms of rationals. There were subsequently rediscovered by J. Sylvester in 1880 and F. Engel in 1913 after whom they are respectively named. A further positive series expansion for real numbers was introduced by J. Lüroth in 1883 and another one in terms of positive rationals by A. Oppenheim in 1972. In 1989 John Peter Louis Knopfmacher and his son Arnold Knopfmacher introduced an algorithm according to which every real number may be expressed by a general alternating series in terms of rationals and found out a unique representation for every real number x of the form x = a0 +

.

 (−1)n−1 1 1 + ... + a1 (a1 + 1)(a2 + 1) . . . (an−1 + 1) an n≥2

in which they gave the name “modified Engel-type alternating series expansion” [29]. The name of “Knopfmacher” in number theory and especially in metrical number theory is connected with the development of a large class of number expansions of series and product type which have revealed important analytic and measure theoretic properties of numbers and of different functionals [22–29]. In parallel it is known that ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems as well. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behaviour of a dynamical system when it is allowed to run for a long time. To this direction more precise information is provided

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by the various ergodic theorems which assert that under certain conditions the time average of a function along the trajectories exist almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff and von Neumann which assert the existence of a time average along each trajectory. The problem of metric classifications of systems is another important part of abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems. Furthermore the concept of ergodicity is central to applications of ergodic theory taking into account that for certain systems the time average of their properties is equal to the average of the entire space. Applications of ergodic theory to other parts of mathematics and especially in metrical number theory usually involve establishing ergodic properties for systems of special kind emphasizing in measure-preserving and ergodic transformations generating number expansions as well as the useful classical criterion for ergodicity of K. Knopp [30] and important properties of measure-preserving, ergodic transformations giving by the well-known Birkhoff-Riesz theorem [21, 36]. All the above applications give the possibility of studying many asymptotic results regarding the digits of number expansions such as the asymptotic frequency of the digits, the arithmetic and geometric means etc. as well as further metrical problems occurring in the context of number theoretic expansions and especially in the context of alternating series and product representations for real numbers [8–11, 14, 15, 17–19, 34, 38, 39, 46]. The main purpose of this work is to bring together the two subjects-alternating number expansions and ergodic theory-by discussing their interconnection as an application of computational mathematics with emphasis on algorithmic number theory and especially on metrical number theory. In particular by considering a general alternating series algorithm that leads to alternating series expansions for real numbers in terms of rationals introduced by A. and J. Knopfmacher the present work arises as an attempt to give some important results arising from the application of ergodic theory to an alternating series expansion for real numbers in terms of rationals arising from another algorithm called alternating Sylvester-Engel-Lüroth series expansion or alternating SEL series expansion which gives generalized versions of the corresponding three alternating series expansions constructing from the general alternating series algorithm of A. and J. Knopfmacher. We focus on alternating representations of the above form for numbers x in .(0, 1) whose elements are random variables defined almost everywhere in .(0, 1) with respect to every probability measure on the .σ -algebra .B(0,1) of all Borel subsets of .(0, 1). In particular we investigate the ergodic behaviour of the basic operator generating the alternating SEL series expansion for any number in .(0, 1) in terms of rationals with respect to the Lebesgue measure .λ. The work is organized as follows. In Sect. 2 we give a brief account of certain concepts of ergodic theory that we shall need throughout the work. In Sect. 3 the general alternating series algorithm introduced by A. and J. Knopfmacher and the algorithm generates the alternating SEL series expansion are considered as well as some important classes of unique alternating series expansions for real numbers in terms of rationals arising from the abovementioned alternating series algorithms. We

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focus on the alternating SEL series expansion for real numbers in order to study the ergodic behaviour of the transformation generating the abovementioned alternating series expansion for any number in .(0, 1) in terms of rationals at it is presented in Sect. 4. Throughout the paper we shall need the following notations: .N = {0, 1, 2, . . .}, ∗ .N = {1, 2, . . .}, .Z = {. . . , −1, 0, 1, . . .}.

2 Preliminaries Definition 1 Let (Ω1 , F1 , P1 ) and (Ω2 , F2 , P2 ) be two probability spaces. (i) A map T : Ω1 → Ω2 is called a transformation. (ii) A transformation T : Ω1 → Ω2 is said to be measurable if for every E ∈ F2 , T −1 E ∈ F1 . (iii) A measurable transformation T : Ω1 → Ω2 is said to be non-singular if for every E ∈ F2 with P2 (E) = 0, P1 (T −1 E) = 0. Definition 2 (i) A transformation T : Ω1 → Ω2 is called measure-preserving (or a homomorphism) if it is measurable and for any E ∈ F2 , P2 (E) = P1 (T −1 E). (ii) A homomorphism is called an isomorphism if T is an one-to-one map of Ω1 onto Ω2 and if T −1 is also a homomorphism. Now let the two abovementioned probability spaces be identical. Definition 3 A measurable nonsingular transformation T is called ergodic if the relation T −1 E = E, for E ∈ F , implies P (E) = 0 or P (E) = 1. If the weaker assumption T −1 E ⊂ E, for E ∈ F , already implies P (E) = 0 or 1, then T is called strongly ergodic. Definition 4 If P1 and P2 are two probability measures on the probability space (Ω, F ), then P1 is said to be absolutely continuous with respect to P2 if P2 (E) = 0 implies P1 (E) = 0 which we denote by P1 0, An

where  An+1 =

.

1 − An · (cn /bn ) with an > 0. an

In the above formula bi = bi (a1 , a2 , . . . ai ), ci = ci (a1 , a2 , . . . , ai )

.

are positive numbers (usually integers), functions of the first i digits .a1 , a2 , . . . ai , 1 chosen so that .An ≤ 1, for .n ∈ N ∗ . Note that .An+1 ≥ 0, since .an ≤ , for An .An > 0.

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3.1.1

Alternating Series Expansions for Real Numbers

By using the abovementioned algorithm we may obtain the following Proposition 1 Every real number x has a unique alternating series representation of the form x = a0 +

.

1 1 1 1 1 · + · − . . . ≡ (a0 , a1 , . . . , an , . . .), − a1 (a1 + 1) a2 (a1 + 1)(a2 + 1) a3

where .an+1 ≥ an , .a1 ≥ 1, for .n ∈ N ∗ . Proof Repeated application of the general alternating series algorithm leads to x = a0 + A1 = a0 +

.

1 b1 − · A2 = a1 c1

b1 · b2 . . . bn−1 1 b1 1 b1 · b2 1 − · + · − . . . + (−1)n−1 · An . a1 c1 a2 c1 · c2 a3 c1 · c2 . . . cn−1

= a0 +

 1 1 1 < An ≤ Since .an = implies that . , for .0 < An ≤ 1, we may take An a1 + 1 an that   cn cn 1 1 1 1 cn · .An+1 = − An · < − = · , if 0 < An ≤ 1. an bn an an+1 bn an (an + 1) bn 

By setting .bn = 1 and .cn = an + 1, for all .n ∈ N ∗ , we may have that  an+1 =

.

1



An+1

≥ an , provided Ai > 0, for i ≤ n.

Furthermore we may take that 1

.

An+1 a ≤ n+1 → 0, as n → ∞, (a1 + 1)(a2 + 1) . . . (an + 1) 2n

since .a1 ≥ 1. Consequently, it follows that every real number x has the following alternating series representation x = a0 +

.

1 1 1 1 1 · · − + − ... a1 a1 + 1 a2 (a1 + 1)(a2 + 1) a3

+ (−1)n−1

1 1 · + ... (a1 + 1)(a2 + 1) . . . (an−1 + 1) an

where .an+1 ≥ an ≥ 1, .n ∈ N ∗ .

(3.1)

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The representation (3.1) is called modified Engel-type alternating expansion for real numbers, while the positive integers .a0 , a1 , . . . , an , . . ., are called digits of the abovementioned expansion. From Proposition 1 we may have the following particular cases (1) x = a0 +

.

1 1 1 + − ... − a1 (a1 + 1)a2 a1 (a1 + 1)a2 (a2 + 1)a3 a1

≡ (a1 , a1 , . . . , an , . . .),

(3.2)

where .an ≥ 1, when we set .cn = an (an + 1), .bn = 1, for all .n ∈ N ∗ . The expansion (3.2) is called Lüroth-type alternating expansion. (2) x = a0 +

.

1 1 1 − + − ... a1 a1 a2 a1 a2 a3

(3.3)

where .an+1 ≥ an + 1, when we set .cn = an , .bn = 1 for all .n ∈ N ∗ . The expansion (3.3) is known as the alternating-Engel expansion. (3) x = a0 +

.

1 1 1 + − ... − a2 a3 a1

(3.4)

where .an+1 ≥ an (an + 1), when we set .cn = bn = 1, for all .n ∈ N ∗ . The expansion (3.4) is known in the literature as the alternating Sylvester expansion. (4) Let x be any real number in the interval .(1, 2). Then regarding alternating product representations for real numbers introduced by A. and J. Knopfmacher x has a unique product expansion of the form    1 (−1)n−1 1 · 1− ... 1 + ... .x = 1+ a2 a2 an

(3.5)

where

an+1 ≥

.

(an + 1)2 ,

if n is odd,

− 1,

if n is even,

an2

an > 1, for every .n ∈ N ∗ . The expansion (3.5) (alternating product expansion) generalizes G. Cantor’s representations by positive products given in 1869. We may focus on any number .x ∗ in the interval .(0, 1). In this case we may have that .x ∗ has a unique

.

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alternating product expansion [11] of the form    1 1 1 1 1 1 · · 1− · − 1+ + 1+ + ... a1 a1 a2 a1 a2 a3     1 (−1)n−2 (−1)n−1 1 · 1− ... 1 + · 1+ + (−1)n+1 · 1 + a1 a2 an−1 an  n 1 (−1) · 1+ + .... (3.6) an+1 an+2

x∗ =

.

(5) Every real number .x > 1 has a unique product expansion of the form x = 2k ·

.

∞   (−1)i−1 1+ ai

(3.7)

i=1



where .ai+1 ≥ (ai + 1) · ai + (−1)i−1 , .k ≥ 0 in Z, regarding the algorithm x = 2k · A∗1 , for 1 ≤ A∗1 < 2,   (−1)n−1 , for n ∈ N ∗ , A∗n /= 1, an = A∗n − 1 −1  (−1)n−1 ∗ An+1 = 1 + · A∗n , n ∈ N ∗ . an .

Furthermore A∗n = 1 + (−1)n−1 · An → 1, as n → +∞

.

and ∗ .A1

=

A∗n+1

∞   (−1)i−1 . 1+ · ai i=1

⨆ ⨅

3.2 Alternating SEL Series Expansion Algorithm We present an algorithm [20, 42] according to which every real number may be expressed by an alternating series expansion in terms of rationals called alternating Sylvester-Engel-Lüroth series expansion or alternating SEL series expansion which gives generalized versions of the above three corresponding alternating series

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expansions arising from the general alternating series algorithm introduced by A. and J. Knopfmacher. (see (3.2), (3.3), (3.4)). Given any real number x, let .a0 = [x], .A1 = x − a0 , .0 ≤ A1 < 1. Then we may recursively define  an =

.

 1 , for An > 0, for any n ∈ N ∗ , An

An+1 = (1 − an An )en ,

(3.8)

where .en = en (an ) is a positive rational number, which may depend on .an . Note that if .0 < An < 1, for any .n ∈ N ∗ then .0 ≤ An+1 < 1 and if .0 < An , ∗ .An+1 < 1 then .an+1 ≥ (an + 1)/en , for any .n ∈ N . Therefore assuming that ∗ ∗ .(an + 1)/en ∈ N , for any .n ∈ N , we take that any real number x may uniquely represented as a series expansion called alternating SEL series expansion of the form x = a0 +

.

 (−1)n 1 + ≡ (a0 , a1 , . . . , an , . . .)SEL , a1 a1 e1 a2 e2 . . . an en an+1 n=1

where .a1 ≥ 1, .an+1 ≥ (an + 1)/en , for any .n ∈ N ∗ . Indeed repeated applications of (3.8) gives 1 A2 1 1 .A1 = − = − a1 a1 e1 a1 a1 e1 =



1 A3 − a2 a2 e2

=

1 1 (−1)n−1 − + ... + a1 a1 e1 a2 a1 e1 a2 e2 . . . an−1 en−1 an +

(−1)n An+1 . a1 e1 a2 e2 . . . an−1 en−1 an en

The process ends in a finite version if some of .An+1 = 0.

4 On the Application of Ergodic Theory to an Alternating SEL Series Expansion Let x be any number in the interval .(0, 1). Then regarding the algorithm (see Sect. 3.2) x has a unique finite or infinite series representation in terms of rationals.

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Indeed let .T : (0, 1) → (0, 1) be the transformation defined by  T (x) =

.

 T n (x) =

1 −x [1/x] , for any x /= 0, 1 [1/x]e1

(4.1)



1

  − T n−1 (x) 1/T n−1 (x) , n ∈ N ∗. 1   1/T n−1 (x) en

If .a1 ≡ a1 (x) = [1/x] and .an+1 ≡ an+1 (x) = [1/(T n (x))], for .T n (x) /= 0 , ∗ .n ∈ N , we obtain that the transformation T defined by (4.1) generates the following unique finite or infinite alternating series representation for any number .x ∈ (0, 1) x=

.

 (−1)n 1 + , a1 a1 e1 a2 e2 . . . an en an+1

(4.2)

n=1

where .a1 ≥ 1, .an+1 ≥ (an + 1)/en , for any .n ∈ N ∗ . Representation (4.2) is called“Sylvester - Engel - Lüroth alternating series expansion, or SEL alternating series expansion” while the positive integers .an ≥ 1, for any .n ∈ N ∗ , are called digits of the abovementioned expansion. We are going to give some important results arising from the application of ergodic theory to theSEL alternating series expansion. We will study the case .en ≥ 1 (positive integer), for any .n ∈ N ∗ . In particular, we find out that the transformation T is ergodic with respect to the Lebesgue measure .λ, but not .λ-measure preserving. To this direction we define by Dn ≡ Dn (i1 , i2 , . . . , in ) = {x ∈ (0, 1)/a1 (x) = i1 , a2 (x) = i2 , . . . , an (x) = in },

.

for any .i1 , . . . , in ∈ N ∗ , the set of all .x ∈ (0, 1) having a unique expansion of the form (4.2) such that the digits .a1 (x), . . . , an (x) take the concrete values .i1 , . . . , in such that .an (x) · en (x)(= in · en ) ≥ 1 (positive integers), .n ∈ N ∗ . Furthermore the set .Dn ≡ Dn (i1 , i2 , . . . , in ) is bounded and its corresponding bounds are given by the relations Mn ≡ sup Dn (i1 , i2 , . . . , in ) =

.

+ (−1)n−1

1 1 1 − + ... i1 i1 e1 i2

1 1 , i1 e1 i2 e2 . . . in−1 en−1 in

On the Application of Ergodic Theory to an Alternating Series Expansion

mn ≡ inf Dn (i1 , i2 , . . . , in ) = + (−1)n−1

95

1 1 1 − + ... i1 i1 e1 i2

1 1 1 + (−1)n , i1 e1 i2 e2 . . . in−1 en−1 in i1 e1 i2 e2 . . . in en

if n is odd. In the case that n is an even number then the above relations are inverted. We consider also the set  1 1 .Drj −1 (x) = x ∈ (0, 1)/rj −1 (x) = − rj (x), rj (x) ∈ (0, 1), ij ij ej  ij ≥ 1, ij ej ≥ 1 (positive integers) , j ∈ N ∗ , where .rj (x) is the jth rest of the SEL alternating series expansion (4.2) defined by rj (x) =

.

1 1 − rj +1 (x), j ∈ N ∗ . aj +1 (x) aj +1 (x)ej +1 (x)

(4.3)

Taking into consideration the probability space .((0, 1), B(0,1) , λ), where .λ is the Lebesgue measure and .B(0,1) the .σ -algebra of all Borel subsets of the interval .(0, 1) we may obtain the following Theorem 3 The rest .rn (x), .n = 0, 1, 2, . . . , of the SEL alternating series expansion satisfy the following relation

λ x ∈ (0, 1)/r0 (x) ∈ Dr0 (x) , r1 (x) ∈ Dr1 (x) , . . . , rn−1 (x) ∈ Drn−1 (x)

.

n 

λ x ∈ (0, 1)/rj −1 (x) ∈ Drj −1 (x) . =

(4.4)

j =1

1 where λ x ∈ (0, 1)/rj −1 (x) ∈ Drj −1 (x) = , j ∈ N ∗. ij ej

(4.5)

Proof In order to prove (4.5) we have that for an arbitrary .j ∈ N ∗ if .aj (x) = ij , then by (4.3) we obtain that rj −1 (x) =

.

1 1 − rj (x), rj (x) ∈ (0, 1), ij ≥ 1, ij ij ej

ij ej ≥ 1 (positive integeres) , j ∈ N ∗ . 1 1 or equivalently .sup Drj −1 (x) = , while if ij ij 1 1 1 1 .rj (x) ≈ 1 , then .rj −1 (x) = − and .inf Drj −1 (x) = − . ij ij ej ij ij ej So if .rj (x) ≈ 0 , then .rj −1 (x) =

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1 Therefore .λ x ∈ (0, 1)/rj −1 (x) ∈ Drj −1 (x) = , j ∈ N ∗. ij ej For the proof of .(4.4) we take the following cases. If (i) n is odd, then

λ x ∈ (0, 1)/r0 (x) ∈ Dr0 (x) , r1 (x) ∈ Dr1 (x) , . . . , rn−1 (x) ∈ Drn−1 (x)  1 1 = λ x ∈ (0, 1)/r0 (x) = − r1 (x), r1 (x) i1 i1 e1 1 1 1 1 = − r2 (x), . . . , rn−1 (x) = − rn (x), rn (x) ∈ (0, 1) i2 i2 e2 in in en  1 1 1 1 1 = λ x ∈ (0, 1)/r0 (x) = − + . . . + (−1)n−1 + i1 i1 e1 i2 i1 e1 i2 e2 . . . in−1 en−1 in 1 rn (x), rn (x) ∈ (0, 1) (−1)n i1 e1 i2 e2 . . . in en

.

= λ (Dn (i1 , i2 , . . . , in )) = Mn − mn =

n 

λ x ∈ (0, 1)/rj −1 (x) ∈ Drj −1 (x) . j =1

We use an analogous argument in the case that (ii) n is an even number. So the proof is complete.

⨆ ⨅

Therefore we are able to prove the following Theorem 4 The operator .T : (0, 1) → (0, 1) defined by (4.1) is ergodic with respect to the Lebesgue measure .λ. Proof At first we define a linear map .gn ≡ gn (i1 , i2 , . . . , in ) : (0, 1) → Dn by gn (t) =

.

n  (−1)j −1 · λ(Dj −1 )

ij

j =1

=

n  j =1

+ (−1)n tλ(Dn )

n  (−1)j −1 1 1 + (−1)n t . i1 e1 i2 e2 . . . ij −1 ej −1 ij ij ej j =1

If .x ∈ Dn then x=

+∞ 

.

j =1

=

(−1)j −1 1 a1 e1 a2 e2 . . . aj −1 ej −1 aj

n  (−1)j −1 λ(Dj −1 )

ij

j =1

= gn (T (x)), n

+ λ(Dn )

+∞  j =n+1

(−1)j −1 1 an+1 en+1 . . . aj −1 ej −1 aj

On the Application of Ergodic Theory to an Alternating Series Expansion

97

which means that .gn−1 = T n : Dn → (0, 1). Moreover .Mn = gn (0), .mn = gn (1), for any .n = 1, 3, 5, . . .. (If n is even then the above relations are inverted). Consequently for any interval .(b1 , b2 ) ⊆ (0, 1), we have λ(T −n (b1 , b2 ) ∩ Dn ) = λ(gn (b1 , b2 ) ∩ Dn ) = |gn (b2 ) − gn (b1 )|

.

= (b2 − b1 ) · λ(Dn ) = λ(b1 , b2 ) · λ(Dn ). Therefore λ(T −n E ∩ Dn ) = λ(E) · λ(Dn ),

.

(4.6)

for any set E in the Boolean ring R of all finite disjoint unions of intervals .(b1 , b2 ) ⊆ (0, 1). It can be easily seen that by using standard measure theory as in [15] the same equation holds for any Borel set E in .(0, 1). Let now E be a Borel set in .(0, 1) such that .T −−1 E = E, where T is a measurable, nonsingular transformation. Then we obtain that .T −−n E = E, for any .n ∈ N ∗ . From relation (4.6) we obtain that .λ(E ∩ Dn ) = λ(E)λ(Dn ) or .λ(E ∩ Dn ) = c · λ(Dn ), with .c = λ(E) > 0. If J is the collection of all cylinders .Dn , .n ∈ N ∗ and .a1 ≥ 1, .aj +1 ≥ (aj + 1)/ej , such that ∗ .ej ≥ 1 with .aj ej ≥ 1 (positive integers), for any .j ∈ N , then any open subinterval of .(0, 1) is an at most denumerable (or countable) disjoint union of elements of J (.λ a.s.). Equivalently we have that .λ(E ∩ B) = c · λ(B), with .c = λ(E) > 0, for any set B which is a countable disjoint union of the fundamental intervals .Dn . Therefore by using Theorem 1 and Definition 3 the proof is complete. ⨆ ⨅ The measurable, non-singular transformation T defined by (4.1) is ergodic but not .λ-measure preserving since for any Borel set E we have that  λ(T

.

−1

−1

E) = λ T

∞ 

E∩

m=1

 D1 (m) =

∞    λ T −1 E ∩ D1 (m) /= λ(E). m=1

Therefore by applying the following form of Birkhoff-Riesz theorem due to Dunford and Miller [5] Theorem 5 Let T be the measurable, non- singular transformation of ((0, 1), B(0,1) , λ) into itself. Assume that there is a constant .M > 0 such that for any Borel set E,

.

−1

n

.

n−1  k=0

λ(T −k E) ≤ Mλ(E), n ∈ N ∗ .

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Then the a.s. limit . lim n−1 n→∞

∞ 

h(T k (x)), for any integrable function h on .(0, 1),

k=0

exists In the study of the metrical properties of the alternating SEL series expansion we are able to obtain important asymptotic results (such as the asymptotic frequency of the digits, the arithmetic and geometric means etc.) which can be studied by using concepts and ideas of a specific theory. Remark It is worth to mention that in [3] it is presented an interconnection between metrical number theory and chaos theory through the application of ergodic theory. In particular by using an additional concept relied on the property of mixing deriving from the Birkoff’s ergodic theorem and the designed Variate Engel Expansion (a variation of the Engel-type expansion for real numbers [8]) the author managed to entangle lossy trapdoor functions from another point of view where the seed of the system became the parameter of the key generation algorithm, while the new designed expansion sequence outputs the one-way trapdoor function and the chaotic behaviour through the ergodic property made the lossy mode accessible. The use of a lossy trapdoor function helps to fix revolving issues surrounding multiparty communications, while the use of Decisional Diffie-Hellman for exchange making a sound proof to what may become a Chosen-Ciphertext Attack resistant scheme if it is studied in the future along distributed systems or adopted in certain communication protocols.

References 1. E. Bach, J. Shallit, Algorithmic Number Theory, Volume I: Efficient Algorithms (Foundations of Computing) (MIT Press, Cambridge, 1996) 2. E. Borel, Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 27, 247–271 (1909) 3. I. Cherkaoui, Diffie-Hellman multi-challenge using a new lossy trapdoor function construction. Int. J. Appl. Math. 51(3), 1–7 (2021) 4. H. Cohen, A Course in Computational Algebraic Number Theory, vol. 138 (Springer, Berlin, 1993) 5. N. Dunford, D.S. Miller, On the ergodic theorem. Trans. Am. Math. Soc. 60, 538–549 (1946) 6. W. Feller, An Introduction to Probability Theory and Its Applications. Vol. I (Wiley, New York, 1968) 7. J. Galambos, Representations of Real Numbers by Infinite Series, vol. 502 (Springer, Berlin, 1976) 8. Ch. Ganatsiou, On the stochastic behaviour of the digits in the modified Engel-type alternating series representations for real numbers, in IBSG Proceedings 5, Proceedings of the Workshop on Global Analysis, Differential Geometry, Lie Algebra’s. Aristotle University of Thessaloniki, July 1997 (Balkan Society of Geometry, Geometry Balkan Press, Bucharest-Romania), pp. 33– 39 9. Ch. Ganatsiou, On some properties of the Lüroth-type alternating series representations for real numbers. Int. J. Math. Math. Sci. 28(6), 367–373 (2001)

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10. Ch. Ganatsiou, On the application of ergodic theory to alternating Engel series. Int. J. Math. Math. Sci. 25(12), 811–817 (2001) 11. Ch. Ganatsiou, K. Perakis, On the ergodicity of an alternating product expansion. Int. J. Pure Appl. Math. 65(3), 381–388 (2010) 12. J.E. Gentle, Foundations of Computational Science (Springer, New York, 2008) 13. A.K. Hartmann, Practical Guide to Computer Simulations (World Scientific, Singapore, 2009) 14. O. Izevblzua, A. Okoromi, On some properties of the alternating Sylvester series and alternating Engel series representations of real numbers. Global J. Math. Sci. 9(1), 53–56 (2010) 15. H. Jager, C. de Vroedt, Lüroth series and their ergodic properties. Nederl. Akad. Wetensch. Proc. Ser. A 31, 31–42 (1969) 16. S. Kalpazidou, Cycle Representations of Markov Processes (Springer, New York, 1995) 17. S. Kalpazidou, Ch. Ganatsiou, Knopfmacher expansions in number theory. Quaest. Math. 24(3), 393–401 (2001) 18. S. Kalpazidou, A. Knopfmacher, J. Knopfmacher, Lüroth-type alternating series representations for real numbers. Acta Arith. 55(4), 311–322 (1990) 19. S. Kalpazidou, A. Knopfmacher, J. Knopfmacher, Metric properties of alternating Lüroth series. Port. Math. 48(3), 319–325 (1991) 20. N.R. Kanasri, P. Singthongla, Alternating SEL series expansion and generalized model construction for the real number system via alternating series. Far East J. Math. Sci. 94(2), 129–147 (2014) 21. M. Kesseböhmer, S. Munday, B.O. Stratmann, Infinite Ergodic Theory of Numbers (De Gruyter, Berlin, 2016) 22. J. Knopfmacher, Ergodic properties of some inverse polynomial series expansions of Laurent series. Acta Math. Hung. 60(3–4), 241–246 (1992) 23. A. Knopfmacher, J. Knopfmacher, A new construction of the real numbers (via infinite products). Nicuw. Arch. Wisk. 25(1), 19–31 (1987) 24. A. Knopfmacher, J. Knopfmacher, A new infinite product representation for real numbers. Mh. Math. 104, 29–44 (1987) 25. A. Knopfmacher, J. Knopfmacher, Inverse polynomial expansions of Laurent series, I–II. Constr. Approx. 4, 379–389 (1988) 26. A. Knopfmacher, J. Knopfmacher, Two concrete new constructions of the real numbers. Rocky Mt. J. Math. 18(4), 813–824 (1988) 27. A. Knopfmacher, J. Knopfmacher, Inverse polynomial expansions of Laurent series, I–II. J. Comput. Appl. Math. 28, 249–257 (1989) 28. A. Knopfmacher, J. Knopfmacher, Representations for real numbers via k-th powers of integers, I–II. Fibonacci Quart. 27, 49–60 (1989) 29. A. Knopfmacher, J. Knopfmacher, Two constructions of the real numbers via alternating series. Int. J. Math. Math. Sci. 12(3), 603–613 (1989) 30. K. Knopp, Mengentheorestische behandlung einiger probleme der diophantischen approximation und der transfiniten wahrscheinlichkeiten. Math. Ann. 95, 409–426 (1926) 31. A.M. Legendre, Essai sur la théorie des nombres 1797-8 (“an vi”), 2nd edn. (1808); 3rd edn. in 2 vol. (1830) 32. T.R. Nonweiler, Computational Mathematics: An Introduction to Numerical Approximation (Wiley, Hoboken, 1984) 33. A. Oppenheim, The representation of real numbers by infinite series of rationals. Acta Arith. 21, 391–398 (1972) 34. M. Pratsiovytyi, Y. Khvorostina, Topological and metric properties of distributions of random variables represented by the alternating Lüroth series with independent elements. Random Oper. Stoch. Equ. 21(4), 385–401 (2013) 35. E.Ya. Remez, On series with alternating sign which may be connected with two algorithms of M.V. Ostrogradskii for approximation of irrational numbers. Uspekhi Mat. Nauk. 6(5(45)), 33–42 (1951) 36. F. Riesz, Sur la theorie ergodique. Comment. Math. Helv. 17, 221–239 (1945)

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37. E. Salamine, Computation of pi using arithmetic–geometric mean. Math. Comput. 30(135), 565–570 (1976) 38. T. Salát, Zur metrischen theorie der Lürothschen Entwicklungen der reellen Zahlen. Czech. Math. J. 18(93), 489–522 (1968) 39. F. Schweiger, Ergodische theorie der Engelschen und Sylvesterschen reihen. Czech. Math. J. 20(95), 243–245 (1970) 40. V. Shoup, A Computational Introduction to Number Theory and Algebra (Cambridge University Press, Cambridge, 2005) 41. W. Sierpinski, Sur quelques algorithmes pour développer les nombres réels en séries. G.R. Soc. Sci. Varsovie 4, 56–77 (1911). French transl. in Oeuvres Choisies, t. I. PWN, Warszawa (1974), 236–254 42. P. Singthongla, N.R. Kanasri, SEL series expansion and generalized model construction for the real number system via series of rationals. Int. J. Math. Math. Sci. 2014, 1–8 (2014) 43. M. Trott, The Mathematica Guidebook for Numerics (Springer, Berlin, 2006) 44. V.A. Uspensky, Complexity and entropy: an introduction to the theory of Kolmogorov complexity, in Kolmogorov Complexity and Its Relations to Computational Complexity Theory, ed. by O. Watanabe (Springer, Berlin, 1992) 45. R.E. White, Computational Mathematics: Models, Methods and Analysis with MATLAB and MPI (Chapman and Hall, Boca Raton, 2015) 46. Yu. Zhykharyeva, M. Pratsiovytyi, Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers. Algebra Discrete Math. 14(1), 145–160 (2012)

Bounds for Similarity Condition Numbers of Unbounded Operators Michael Gil’

AMS (MOS) Subject Classification 47B40, 47E05, 47A30, 47A55, 47A56

1 Introduction Let .H √ be a complex separable Hilbert space with a scalar product .(., .), the norm ‖.‖ = (., .) and unit operator I . For a linear operator A on .H , .Dom(A) is the domain, .A∗ is the operator adjoint to A; .σ (A) denotes the spectrum of A and .A−1 is the inverse one to A; .Rλ (A) = (A − I λ)−1 .(λ /∈ σ (A)) is the resolvent; .AI := (A − A∗ )/2i; .λj (A) .(j = 1, 2, . . .) are the eigenvalues of A taken with their multiplicities and enumerated as .|λj (A)| ≤ |λj +1 (A)|. In addition, .ρ(A, λ) = infs∈σ (A) |s − λ| .(λ ∈ C). So .ρ(A, λ) is the distance between .σ (A) and a complex point .λ. By .Sp .(1 ≤ p < ∞) we denote the Schatten-von Neumann ideal of compact operators K with the finite norm .Np (K) := [trace (KK ∗ )p/2 ]1/p . In particular, .S2 is the Hilbert-Schmidt ideal. Two operators A and .A˜ are said to be similar if there exists a boundedly invertible ˜ = T −1 AT x .(x ∈ Dom(A)). ˜ The constant .κT := bounded operator T such that .Ax −1 ‖T ‖‖T ‖ is called the condition number. Conditions that provide the similarity of various operators to normal, selfadjoint and other types of operators were considered by many mathematicians, cf. [2, 5, 6, 11, 12, 27–29, 31, 34], and references given therein. In particular, in the paper [7] condition number estimates are suggested for combined potential boundary integral operators in acoustic scattering; in the paper [37] condition numbers are estimated for second-order elliptic operators. The generalizations of condition numbers of bounded linear operators in Banach spaces were explored in the paper [8]. It should be noted that in many cases, the condition numbers are numerically calculated, e.g. [3, 33]. In the present paper for some new classes of unbounded operators in .H we .

M. Gil’ () Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_7

101

102

M. Gil’

prove their similarity to normal operators and derive sharp bounds for the condition numbers. Illustrative examples of non-selfadjoint differential operators and integrodifferential operators are also presented. In addition, we discuss applications of the condition numbers to spectrum perturbations and operator functions. We will say that a discrete spectrum of an operator is algebraically simple, if the algebraic multiplicity of each its eigenvalue is equal to one, i.e. the eigen-space corresponding to each eigenvalue is one-dimensional. A few words about the contents. The paper consists of 14 sections. Let H be a linear operator on .H . In Sects. 2–5 it is supposed that .Dom(H ) = Dom(H ∗ ), H has a compact resolvent and a Hilbert-Schmidt imaginary Hermitian component .HI = (H − H ∗ )/2i. In addition, .σ (H ) is algebraically simple. In Sects. 6–8 we investigate the operator H having a compact resolvent and a Schatten-von Neumann Hermitian component. Numerous integro-differential operators are examples of operators considered in Sects. 2–8. Sections 9–11 deal with the operator H satisfying the following condition: there is a normal operator D with a discrete algebraically simple spectrum and, in addition, the norm .‖H − D‖ is finite and sufficiently small. In Sect. 12 we consider the operator H assuming that there is an invertible selfadjoint operator S with a discrete algebraically simple spectrum, and .‖(H − S)S −ν ‖ < ∞ for a .ν ∈ [0, 1). In Sect. 13 we discuss applications of the condition numbers to spectrum perturbations and operator functions. Section 14 contains bibliographical comments.

2 Operators with Hilbert-Schmidt Hermitian Components Let H be a linear operator on .H with a compact inverse one, algebraically simple spectrum and .Dom(H ) = Dom(H ∗ ). In addition, HI := (H − H ∗ )/2i ∈ S2

.

(1)

For a fixed integer m put δm (H ) =

.

inf

j =1,2,...; j /=m

|λj (H ) − λm (H )|.

It is further supposed that ⎡ ∞  .ζ (H ) := ⎣ j =1

⎤1/2 1 ⎦ < ∞. δj2 (H )

(2)

Similarity Condition Numbers

103

Hence it follows that ˆ ) := inf δm (H ) = δ(H

.

m

inf

j /=k;j,k=1,2,...

|λj (H ) − λk (H )| > 0.

Denote ∞ √  ( 2N2 (HI ))k+1 .τ (H ) := and γ (H ) := exp [2ζ (H )τ (H )]. √ k!δˆk (H ) k=0

It follows from condition (2) that .δj (H ) ∼ j α+1/2 for some .α > 0. That is, .δj (H ) increases more rapidly than .j 1/2 . So we can interpret this condition to mean that the eigenvalues of H are in some sense widely separated. Theorem 1 Let conditions (1) and (2) be fulfilled, and .σ (H ) be algebraically simple. Then there are a boundedly invertible bounded operator T and a normal operator D acting in .H , such that T H x = DT x (x ∈ Dom(H )).

.

Moreover, κT = ‖T −1 ‖‖T ‖ ≤ γ (H ).

.

The proof of this theorem is divided into a series of lemmas which are presented in the next three sections. The theorem is sharp: if H is selfadjoint, then .γ (H ) = 1. To illustrate Theorem 1, put .HR = (A + A∗ )/2 and suppose that λj +1 (HR ) − λj (HR ) ≥ b0 j α (b0 = const > 0; α > 1/2; j = 1, 2, . . .).

.

(3)

It can be directly checked that the condition .‖Rλ (HR )‖‖HI ‖ < 1 implies .λ /∈ σ (H ). Since .HR is selfadjoint, we have .‖Rλ (HR )‖ = ρ −1 (HR , λ), and .λ /∈ σ (H ), provided .‖HI ‖ < ρ(HR , λ). Hence .‖HI ‖ ≥ ρ(HR , μ) for any .μ ∈ σ (H ). This implies the relation .

sup inf |λk (H ) − λj (HR )| ≤ ‖HI ‖. k

j

Thus, if .σ (HR ) is simple and 2‖HI ‖ < inf(λj +1 (HR ) − λj (HR )),

.

j

104

M. Gil’

ˆ ) ≥ infj (λj +1 (HR ) − λj (HR ) − 2‖HI ‖) and according then .σ (H ) is simple, .δ(H to (3), condition (2) holds with ζ (H ) ≤

.

∞  (λj +1 (HR ) − λj (HR ) − 2‖HI ‖)−2 < ∞.

(4)

j =1

Example 1 Consider in space .L2 (0, 1) the operator H defined by H = −d 2 /dx 2 + iK

.

with Dom (H ) = {v ∈ L2 (0, 1) : v '' ∈ L2 (0, 1), v(0) = v(1) = 0},

.

where K is a self-adjoint Hilbert-Schmidt operator. Take .HR = −d 2 /dx 2 with .Dom (HR ) = Dom (H ). Then .λj (HR ) = π 2 j 2 (j = ˆ )≥ 1, 2, . . .) and .λj +1 (HR ) − λj (HR ) = π 2 (2j + 1). So, if .2‖K‖ < 3π 2 , then .δ(H 3π 2 − 2‖K‖ and due to (4) ζ (H ) ≤

.

∞  (π 2 (2j + 1) − 2‖K‖)−2 < ∞. j =1

Now one can directly apply Theorem 1.

3 The Sylvester Equation Let .B0 be a bounded linear operator in .H having a finite chain of invariant projections .Pk .(k = 1, . . . , n; n < ∞): 0 ⊂ P1 H ⊂ P2 H ⊂ . . . ⊂ Pn H = H

(5)

Pk B0 Pk = B0 Pk (k = 1, . . . , n).

(6)

.

and .

Let .H1 ⊂ H be an invariant subspace of an operator A. Then .A|H1 means the restriction of A onto .H1 .

Similarity Condition Numbers

105

Put ΔPk = Pk − Pk−1 (P0 = 0), Aˆ k = ΔPk B0 ΔPk and Ak = Aˆ k |ΔPk H .

.

Then .σ (Aˆ k ) = σ (Ak ) ∪ {0}. Lemma 1 One has σ (B0 ) = ∪nk=1 σ (Ak ).

.

Proof Put Dˆ =

n 

.

ˆ Ak and W = B0 − D.

k=1

Due to (6) we have .W Pk = Pk−1 W Pk . Hence, W n = W n Pn = W n−1 Pn−1 W Pn = W n−2 Pn−2 W Pn−1 W Pn =

.

W n−2 Pn−2 W 2 = W n−3 Pn−3 W 3 = . . . = P0 W n = 0.

.

So W is nilpotent. Similarly, taking into account that (Dˆ − λI )−1 W Pk = (Dˆ − λI )−1 Pk−1 W Pk = Pk−1 (Dˆ − λI )−1 W Pk

.

we prove that .((Dˆ − λI )−1 W )n = 0 .(λ /∈ σ (D)). Thus (B0 − λI )−1 = (Dˆ + W − λI )−1 = (I + (Dˆ − λI )−1 W )−1 (Dˆ − λI )−1 =

.

.

n−1  (−1)k ((Dˆ − λI )−1 W )k (Dˆ − λI )−1 . k=0

ˆ = σ (B0 ). Since .Ak are mutually orthogonal this Hence it easily follows that .σ (D) proves the lemma. ⨆ ⨅ Taking into account (5) and (6) put Qk = I − Pk , Bk = Qk B0 Qk and Ck = ΔPk B0 Qk (k = 1, . . . , n − 1).

.

and assume that σ (Ak ) ∩ σ (Aj ) = ∅ (j /= k; j, k = 1, . . . , n).

.

(7)

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M. Gil’

Since .Bj is a block triangular operator matrix, according to the previous lemma we have σ (Bj ) = ∪nk=j +1 σ (Ak ) (j = 0, . . . , n − 1).

.

(8)

We need the following result. Theorem 2 (Rosenblum [36]) Let .H be a Hilbert space and let .A, B, Y be bounded linear operators on .H . Suppose that .σ (A) ∩ σ (B) = ∅. Under these conditions the operator equation .AX − XB + Y = 0 has a unique solution X given by X=

.

1 2π i



(zI − A)−1 Y (zI − B)−1 dz,

Γ

where .Γ is a piecewise smooth closed curve with .σ (A) outside .Γ and .σ (B) inside Γ.

.

Due to (7) and (8) σ (Bj ) ∩ σ (Aj ) = ∅ (j = 1, . . . , n).

.

Under this condition, according to the Rosenblum theorem, the equation Aj Xj − Xj Bj = −Cj (j = 1, . . . , n − 1)

.

has a unique solution Xj : Qj H → ΔPj H .

.

(see also [9, Section I.3] and [4]). Besides, we can write .Xj = ΔPj Xj Qj . Lemma 2 Let condition (7) hold and .Xj be a solution to (9). Then (I − Xn−1 )(I − Xn−2 ) · · · (I − X1 ) B0 (I + X1 )(I + X2 ) · · · (I + Xn−1 ) =

.

ˆ A1 + A2 + . . . + An = D.

.

Proof Since .Xj = ΔPj Xj Qj , we have Xj Aj = Bj Xj = Xj Cj = Cj Xj = 0.

.

Clearly, .Qj B0 Pj = 0. Thus, .B0 = A1 + B1 + C1 and consequently, (I − X1 )B0 (I + X1 ) = (I − X1 )(A1 + B1 + C1 )(I + X1 ) =

.

A1 + B1 + C1 − X1 B1 + A1 X1 = A1 + B1 .

.

(9)

Similarity Condition Numbers

107

Furthermore, .B1 = A2 + B2 + C2 . Hence, (Q1 − X2 )B1 (Q1 + X2 ) = (Q1 − X1 )(A2 + B2 + C2 )(Q1 + X1 ) =

.

A2 + B2 + C2 − X2 B2 + A2 X2 = A2 + B2 .

.

Therefore, (I − X2 )(A1 + B1 )(I + X2 ) = (P1 + Q1 − X2 )(A1 + B1 )(P1 + Q1 + X2 ) =

.

A1 + (Q1 − X2 )(A1 + B1 )(Q1 + X2 ) = A1 + A2 + B2 .

.

Consequently, (I − X2 )(A1 + B1 )(I + X2 ) = (I − X2 )(I − X1 )B0 (I + X1 )(I + X2 ) =

.

A1 + A2 + B2 .

.

Continuing this process and taking into account that .Bn−1 = An , we obtain the required result. ⨅ ⨆ Take Tˆn = (I + X1 )(I + X2 ) · · · (I + Xn−1 ).

.

(10)

Since .Xj = ΔPj Xj Qj , we obtain .Xj2 = 0. Hence .(I + Xj )(I − Xj ) = (I − Xj )(I + Xj ) = I , and the inverse to .I + Xj is the operator .I − Xj . Thus, Tˆn−1 = (I − Xn−1 )(I − Xn−2 ) · · · (I − X1 )

.

(11)

and due to Lemma 2, Tˆn−1 B0 Tˆn = diag (Ak )nk=1 .

.

(12)

By the inequalities between the arithmetic and geometric means we get ‖Tˆn ‖ ≤

n−1 n−1 1  (1 + ‖Xk ‖) ≤ 1 + ‖Xk ‖ n−1

n−1 

.

k=1

(13)

k=1

and

−1 .‖Tˆn ‖

n−1 n−1 1  ≤ 1+ ‖Xk ‖ . n−1 k=1

(14)

108

M. Gil’

4 The Finite Dimensional Case Let .Cn×n be the set of .n × n-matrices and .A ∈ Cn×n has an algebraically simple spectrum. So ˆ δ(A) :=

.

min

j,k=1,...,n; k/=j

|λj (A) − λk (A)| > 0.

Put g(A) = [N22 (A) −

n 

.

|λk (A)|2 ]1/2 .

k=1

As it is shown in [22, Theorem 3.1] , g(A) = [2N22 (AI ) − 2

n 

.

|𝔍 λk (A)|2 ]1/2 ≤

√

2N2 (AI ) (AI = (A − A∗ )/2i).

k=1

(15) Furthermore, for a fixed .m ≤ n put δm (A) =

.

min

j =1,2,...,n; j /=m

|λj (A) − λm (A)|, ζˆn (A) =

n−1  k=1

τˆn (A) :=

.

1 2 δk (A)

1/2 ,

n−1  g k+1 (A) √ k!δˆk (A) k=0

and

ζˆn (A)τˆn (A) .γˆn (A) := 1+ n−1

2(n−1) .

We need the following result. Lemma 3 Let the eigenvalues of .A ∈ Cn×n be algebraically simple. Then there is an invertible operator .Tn and a normal operator .Dn , such that the equality Tn−1 ATn = Dn

.

holds with .κTn := ‖Tn−1 ‖‖Tn ‖ ≤ γˆn (A).

(16)

Similarity Condition Numbers

109

Proof Let .{ek } be the Schur basis (the orthogonal normal basis of the triangular representation) of matrix A: ⎛

a11 ⎜ 0 .A = ⎜ ⎝ . 0

a12 a22 . 0

a13 a23 . 0

... ... ... ...

⎞ a1n a2n ⎟ ⎟ . ⎠ ann

with .ajj = λj (A). Besides, k−1 n   .

|aj k |2 = g 2 (A),

k=2 j =1

(see also [14, Lemma 2.3.2]). To apply Lemma 2 take .Pj = .ΔPk = (., ek )ek , Qj =

n 

.

j

k=1 (., ek )ek , .B0

= A,

(., ek )ek , Ak = ΔPk AΔPk = λk (A)ΔPk ,

k=j +1

⎛

aj +1,j +1 aj +1,j +2 ⎜ 0 aj +2,j +2 .Bj = Qj AQj = ⎜ ⎝ . . 0 0

⎞ . . . aj +1,n . . . aj +2,n ⎟ ⎟, . ... ⎠ . ann

  Cj = ΔPj AQj == aj,j +1 aj,j +2 . . . aj,n

.

and Dn = diag(λk (A)).

.

In addition,  A=

.

λ1 (A) C1 0 B1



 , B1 =

λ2 (A) C2 0 B2



 , . . . , Bj =

λj +1 (A) Cj +1 0 Bj +1



(j < n). So .Bj is an upper-triangular .(n − j ) × (n − j )-matrix. Equation (9) takes the form

.

λj (A)Xj − Xj Bj = −Cj .

.

110

M. Gil’

Since .Xj = Xj Qj , we can write .Xj (λj (A)Qj − Bj ) = −Cj . Therefore, Xj = −Cj (λj (A)Qj − Bj )−1 .

.

The inverse operator is understood in the sense of subspace .Qj Cn . Hence, ‖Xj ‖ ≤ ‖Cj ‖‖(λj (A)Qj − Bj )−1 ‖.

.

Take .Tn = Tˆn as in (10) with .Xk = −Ck (λk (A)Qk − Bk )−1 . By Lemma 2 and (12) Tn−1 ATn = diag (Ak )nk=1 ,

.

Besides, according to (13) and (14) ‖Tn ‖ ≤

n−1 

(1 +

.

k=1

1 ‖Ck (λk (A)Qk − Bk )−1 ‖) n−1

(17)

1 ‖Ck (λk (A)Qk − Bk )−1 ‖). n−1

(18)

and −1 .‖Tn ‖

≤

n−1 

(1 +

k=1

Besides, n 

‖Cj ‖2 =

|aj k |2

.

k=j +1

and due to [14, Corollary 2.1.2], we have ‖(λj (A)Qj − Bj )−1 ‖ ≤

n−j −1

.

k=0

√

g k (Bj ) k!δjk+1 (A)

.

Since .g(A) is equal to the Hilbert-Schmidt norm of the nilpotent part of A (see [22, Section 3.1]) we obtain .g(Bj ) = g(Qj AQj ) ≤ g(A) .(j ≥ 1). So ‖(λj (A)Qj − Bj )−1 ‖ ≤

n−1 

.

k=0

√

g k (A) k!δjk+1 (A)

≤

n−1  k=0

g k (A) τˆn (A) = √ k g(A)δj (A) k!δj (A)δˆ (A)

Similarity Condition Numbers

111

and thus, ‖Xj ‖ ≤

.

‖Cj ‖τˆn (A) . g(A)δj (A)

Besides (13) and (14) imply ⎞n−1 ⎛ ⎞n−1 n−1 n−1   ‖C ‖ (A) τ ˆ 1 j ⎠ n .‖Tn ‖ ≤ ⎝1 + ‖Xj ||⎠ ≤ ⎝1 + g(A)(n − 1) δj (A) n−1 ⎛

j =1

j =1

and ⎛

⎞n−1 n−1  ‖C ‖ (A) τ ˆ j n −1 ⎠ . .‖Tn ‖ ≤ ⎝1 + g(A)(n − 1) δj (A) j =1

But by the Schwarz inequality, (

.

n−1 n−1 n−1   ‖Cj ‖ 2  1 ) ≤ . ‖Cj ‖2 2 δj (A) δk (A) j =1

j =1

k=1

In addition, n−1  .

‖Cj ‖2 ≤

j =1

n n−1  

|aj k |2 = g 2 (A).

j =1 k=j +1

Thus .‖Tn ‖2 ≤ γˆn (A) and .‖Tn−1 ‖2 ≤ γˆn (A). This proves the lemma.

⨆ ⨅

5 Proof of Theorem 1 We need the following result. Lemma 4 Let a linear operator H on .H have a compact resolvent, and for some b /∈ σ (H ), .(H − bI )−1 have a complete system of root vectors. Then there is an orthogonal normal (Schur) basis .{ek }∞ k=1 , in which H is representable by a triangular matrix .(aj k )1≤j ≤k≤∞ :

.

H ek =

k 

.

aj k ej and (H ek , ek ) = λk (λk = λk (H ), k = 1, 2, . . .),

j =1

where .λk (H ) (k = 1, 2, . . .) are the eigenvalues of H .

(19)

112

M. Gil’

For the proof see [25, Lemma 2.4]. Proof of Theorem 1 Recall that .σ (H ) is algebraically simple. Due to [26, Theorem V.10.1], the system of eigenvectors of H is complete in .H . But the eigenvectors of H and .H −1 coincide. From the previous lemma it follows that there isan orthogonal k ˆ normal (Schur) basis .{eˆk }∞ j =1 (., eˆj )eˆj . k=1 , such that (19) holds. Denote .Pk = Then H Pˆk = Pˆk H Pˆk (k = 1, 2, . . .).

.

(20)

Besides, ΔPˆk H ΔPˆk = λk ΔPˆk (ΔPˆk = Pˆk − Pˆk−1 , k = 1, 2, . . . ; Pˆ0 = 0).

.

Put D=

∞ 

.

λk (H )ΔPˆk and V = H − D.

k=1

Furthermore, put .Hn = H Pn . Due to (20) we have ‖Hn f − Hf ‖ → 0 (f ∈ Dom(H )) as n → ∞.

.

From Lemma 3 with .A = Hn it follows that in .Pˆn H there is a invertible operator Tn such that .Tn Hn = Pˆn DTn and .‖Tn ‖2 ≤ γˆn (Hn ) ≤ γ (H ). So there is a weakly convergent subsequence .Tnj whose limit we denote by T . It is simple to check that .Tn = Pn T . So in fact the pointed subsequence converges strongly. Thus .Tnj Hnj f → T Hf and, therefore .Pˆnj DTnj f = Tnj Hnj f → T Hf . Letting .nj → ∞ hence we arrive at the required result. ⨆ ⨅ .

6

Operators with Schatten-von Neumann Hermitian Components

Let H be a linear operator on .H , with a compact inverse one, .Dom(H ) = Dom(H ∗ ) and HI = (H − H ∗ )/2i ∈ S2p for an integer p ≥ 2.

.

Again assume that .σ (H ) is algebraically simple and put δm (H ) =

.

inf

j =1,2,...; j /=m

|λj (H ) − λm (H )| (m = 1, 2, . . .).

(21)

Similarity Condition Numbers

113

It is further supposed that ⎡ ζp (H ) := ⎣

∞ 

.

j =1

⎤1/q 1 1 ⎦ 1 = 1). 0.

Denote p−1 ∞ β  √ p 2ζp (H )

kp+m

up (H ) :=

.

m=0 k=0

kp+m+1

N2p

(HI ) , √ δˆkp+m (H ) k!

where 

 2p .βp := 2 1 + . e2/3 ln2 Theorem 3 Let conditions (21) and (22) be fulfilled, and .σ (H ) be algebraically simple. Then there are a boundedly invertible bounded operator T and a normal operator D acting in .H , such that T H x = DT x (x ∈ Dom(H ))

(23)

κT = ‖T −1 ‖‖T ‖ ≤ e2up (H )

(24)

.

and .

The proof of this theorem is presented in the next two sections. The theorem is sharp: if H is selfadjoint, then .up (H ) = 0 and we obtain .κT = 1. We will show that one can replace (24) by the inequality κT ≤ e2uˆ p (H ) ,

(25)

.

where m+1 p−1  βpm N2p (HI ) √ .u ˆ p (H ) := 2e ζp (H ) exp m δˆ (A) m=0



 (βp N2p (HI ))2p . 2δˆ2p (A)

114

M. Gil’

Example 2 Consider in .L2 (0, 1) the spectral problem u(4) (x) + (iKu)(x) = λu(x) (λ ∈ C, 0 < x < 1);

.

u(0) = u(1) = u'' (0) = u'' (1) = 0, where .K ∈ S2p is selfadjoint. Take .H = d 4 /dx 4 + iK with Dom (H ) = {v ∈ L2 (0, 1) : v (4) ∈ L2 (0, 1), v(0) = v(1) = v '' (0) = v '' (1) = 0}.

.

Then .HR = d 4 /dx 4 with .Dom (HR ) = Dom (H ), .λj (HR ) = π 4 j 4 (j = 1, 2, . . .) ˆ ) ≥ 4π 4 − 2‖K‖ and and .λj +1 (HR ) − λj (HR ) ≥ 4π 4 j 3 . If .‖K‖ < 2π 4 , then .δ(H q

ζp (H ) ≤

.

∞  (4π 4 j 3 − 2‖K‖)−q < ∞. j =1

Now one can directly apply Theorem 3.

7 A Generalization of Lemma 3 To prove Theorem 3 we need the following result. Theorem 4 Let M be a linear operator on .H , such that .Dom (M) = Dom (M ∗ ) and .MI = (M − M ∗ )/2i ∈ S2p for some integer .p ≥ 2. Then p−1 ∞ 

‖Rλ (M)‖ ≤

.

m=0 k=0

(βp N2p (MI ))kp+m (λ /∈ σ (M)). √ ρ pk+m+1 (M, λ) k!

(26)

Moreover, one has √  (βp N2p (MI ))m exp e .‖Rλ (M)‖ ≤ ρ m+1 (M, λ) p−1

m=0



(βp N2p (MI ))2p 2ρ 2p (M, λ)

 (λ /∈ σ (M)). (27)

For the proof see [14, Theorem 7.9.1], [22, Theorem 11.1]. In the rest of this section A is an .n × n-matrix with an algebraically simple spectrum. So A is diagonalizable: there is an invertible matrix .Tn ∈ Cn×n and a normal matrix .Dn ∈ Cn×n , such that .Tn−1 ATn = Dn . Moreover, one can take n . .Dn = diag (λk (A)) k=1

Similarity Condition Numbers

115

ˆ Define .δ(A) as in Sect. 4, and for a fixed .j ≤ n again put δj (A) =

.

inf

m=1,2,...,n; m/=j

|λj (A) − λm (A)|

Besides, according to (17) and (18) ‖Tn ‖ ≤

n−1 

(1 +

.

k=1

1 ‖Ck (λk (A)Qk − Bk )−1 ‖) n−1

(28)

1 ‖Ck (λk (A)Qk − Bk )−1 ‖), n−1

(29)

and ‖Tn−1 ‖ ≤

n−1 

(1 +

.

k=1

where .Ck , Qk , Bk are defined in Sect. 4. Due to (26) ‖(λj (A)Qj − Bj )−1 ‖ ≤

p−1 ∞ 

.

m=0 k=0

(βp N2p (Bj I )kp+m √ , kp+m+1 δj (A) k!

where .Bj I is the imaginary Hermitian component of .Bj . But .N2p (Bj I ) = N2p (Qj AI Qj ) ≤ N2p (AI ) .(j ≥ 1). So ‖(λj (A)Qj − Bj )−1 ‖ ≤

.

τ (A) δj (A)

where τ (A) =

p−1 ∞ 

.

m=0 k=0

(βp N2p (AI ))kp+m . √ δˆkp+m (A) k!

Besides (28) and (29) imply ⎛

⎞n−1 n−1  ‖Cj ‖ τ (A) ⎠ .‖Tn ‖ ≤ ⎝1 + (n − 1) δj (A) j =1

and ⎛

⎞n−1 n−1  ‖C ‖ τ (A) j ⎠ −1 .‖Tn ‖ ≤ ⎝1 + . δj (A) (n − 1) j =1

(30)

116

M. Gil’

But by the Hólder inequality, ⎛ ⎞1/2p n−1 n−1   ‖Cj ‖ ≤⎝ ‖Cj ‖2p ⎠ ζp (A) (1/(2p) + 1/q = 1), . δj (A)

(31)

j =1

j =1

where ζp (A) :=

n−1 

.

k=1

1 q δk (A)

1/q .

In addition, according to the definition of .Cj , n 

‖Cj ‖2 ≤

.

|aj k |2 , j < n; Cn = 0,

k=j +1

and n 

4‖AI ej ‖2 = ‖(A − A∗ )ej ‖2 = |ajj − a jj |2 + 2

.

|aj k |2 ≥ 2‖Cj ‖2 ; j < n.

k=k=j +1

Recall that .{ek } is the Schur basis of A. Thus, .‖Cj ‖ ≤ therefore n−1  .

‖Cj ‖

2p

j =1

≤2

p

n−1 

√ 2‖AI ej ‖, j ≤ n and

‖AI ej ‖2p .

j =1

But from Lemmas II.4.1 and II.8.4 [26], it follows that n−1  .

2p

‖AI ej ‖2p ≤ N2p (AI ).

j =1

Therefore relations (30), (31) with the notation

n−1 √ τ (A) 2N2p (AI )ζp (A) .ψn,p (A) = 1+ n−1 imply .‖Tn ‖ ≤ ψn,p (A) and .‖Tn−1 ‖ ≤ ψn,p (A). We thus have proved the following.

Similarity Condition Numbers

117

Lemma 5 Let .A ∈ Cn×n have an algebraically simple spectrum. Then there is an invertible operator .Tn ∈ Cn×n , such that (16) holds with .κTn = ‖Tn−1 ‖‖Tn ‖ ≤ 2 (A). ψn,p

8 Proof of Theorem 3 Due to Lemma 4, under the hypothesis of Theorem 3, there is an orthonormal (Schur) basis .{eˆk }∞ k=1 , in which H is represented by a triangular matrix. We have H Pˆk f = Pˆk H Pˆk f (k = 1, 2, . . . ; f ∈ Dom(H )).

.

where .Pˆk = D=

k

∞ 

.

j =1 (., eˆj )eˆj .

(32)

Put

λk ΔPˆk (ΔPˆk = Pˆk − Pˆk−1 , k = 1, 2, . . .) and V = H − D.

k=1

Furthermore, put .Hn = H Pn . Due to (32) we have ‖Hn f − Hf ‖ → 0 (f ∈ Dom(H )) as n → ∞.

.

(33)

From Lemma 5 and (33) with .A = Hn it follows that in .Pˆn H there is a invertible operator .Tn such that .Tn Hn = Pˆn DTn and √ τ (Hn ) 2N2p (HnI )ζp (Hn ) n−1 ) .‖Tn ‖ ≤ ψn,p (Hn ) := (1 + n−1 where τ (Hn ) =

p−1 ∞ 

.

m=0 k=0

(βp N2p (HnI ))kp+m . √ δˆkp+m (Hn ) k!

It is clear, that √ √ τ (Hn ) 2N2p (HnI )ζp (Hn ) ≤ τ (H ) 2N2p (HI )ζp (H ) = up (H )

.

and therefore ‖Tn ‖ ≤ (1 +

.

Similarly, .‖Tn−1 ‖ ≤ eup (H ) .

up (H ) n−1 ) ≤ eup (H ) . n−1

118

M. Gil’

So there is a weakly convergent subsequence .Tnj whose limit we denote by T . It is simple to check that .Tn = Pn T . Since projections .Pn converge strongly, subsequence .{Tnj } converges strongly. Thus .Tnj Hnj f → T Hf strongly and, therefore .Pˆnj DTnj f = Tnj Hnj f → T Hf strongly. Letting .nj → ∞ hence we arrive at the required result. ⨆ ⨅ Note that to prove (25) in our arguments one can apply (27) instead of (26).

9 Operators “Close” to Normal Ones In the present section we estimate the condition number of a linear operator A on .H with the following property: there is a normal operator D with a discrete spectrum, such that .Dom (A) = Dom (D), and ς := ‖A − D‖ < ∞.

(34)

.

It is assumed .σ (D) is algebraically simple: .infj =1,2,...; j /=m |λj (D)−λm (D)|/2 > 0 (m = 1, 2, . . .) and

.

dˆ := inf dm > 0.

.

(35)

m

Theorem 5 Let the conditions (34), (35), 2ς < dˆ

(36)

.

and ∞  .

k=1

1 0). Under condition (40), let .A1 have an eigenvalue .λˆ (A1 ) and d :=

.

1 := inf |λj (A) − λˆ (A1 )| > 0. 2 j =1,2,...; λj (A1 )/=λˆ (A1 )

(41)

Suppose that qφ(1/d) < 1.

(42)

.

Since .Rλ (A1 ) − Rλ (A2 ) = Rλ (A1 )(A2 − A1 )Rλ (A2 ), from (40) and (42) it follows that ‖Rλ (A2 )‖ ≤

.

φ(1/d) ‖Rλ (A1 )‖ ≤ < ∞ (λ ∈ ∂Ω(λ(A1 ), d)). 1 − qφ(1/d) 1 − qφ(1/d)

Put 1 .P (A1 ) = − 2π i

 |λˆ (A1 )−λ|=d

and P (A2 ) = −

1 2π i

Rλ (A1 )dλ



|λˆ (A1 )−λ|=d

Rλ (A2 )dλ.

That is, .P (A1 ) and .P (A2 ) are the Riesz projections onto the eigenspaces of .A1 and .A2 , respectively, corresponding to the points of the spectra, which belong to .Ω(λ(A1 ), d). Lemma 6 Let .A1 satisfy condition (40) and have an eigenvalue .λˆ (A1 ) of the algebraic multiplicity .μ. If, in addition, the condition qφ(1/d)[1 + φ(1/d)d] < 1

.

(43)

Similarity Condition Numbers

121

holds, where d is defined by (41), then .dim P (A1 )H = dim P (A2 )H = μ and ‖P (A1 ) − P (A2 )‖ ≤ δ, where δ :=

.

qdφ 2 (1/d) < 1. 1 − qφ(1/d)

The proofs of this lemma and of the next one can be found in [13, Lemma 2.1] and [13, Theorem 1.1], respectively. Lemma 7 Suppose .A1 has an eigenvalue .λˆ (A1 ) whose algebraic multiplicity is equal to one and conditions (40) and (43) hold. Then .A2 has in .Ω(λˆ (A1 ), d) an eigenvalue .λ(A2 ) whose algebraic multiplicity is equal to one. Moreover, the normed eigenvectors .e(A2 ) and .e(A1 ) corresponding to .λ(A2 ) and .λ(A1 ) satisfy the inequality ‖e(A2 ) − e(A1 )‖ ≤

.

2δ . 1−δ

11 Proof of Theorem 5 Let .{ek } be the set of all (orthogonal) normed eigenvectors of D. So D=

∞ 

.

λk (D)Pk , where Pk = (., ek )ek .

k=1

Denote by .Qm the Riesz projection of A corresponding to the eigenvalues of A lying in .Ω(λm (D), dm ) and suppose that (34) and (35) hold. Since D is normal, we have −1 (D, λ). Thus .λ /∈ σ (A), provided .ς < ρ(D, λ). Hence it easily .‖Rλ (D)‖ = ρ follows that .

sup

inf |s − t| ≤ ς.

s∈σ (A) t∈σ (D)

In the selfadjoint case this inequality is given in [30, p. 291]. Consequently, σ (A) ⊂ ∪∞ m=1 Ω(λm (D), ς ).

.

Use Lemma 6 with .A1 = D, .A2 = A. In this case .φ(x) = x, .‖Rλ (D)‖ ≤ 1/ρ(D, λ). Under condition (36), inequality (43) holds. Thus due to that lemma .Qm is one-dimensional and ‖Pm − Qm ‖ ≤ δm , where δm :=

.

ς < 1. dm − ς

(44)

122

M. Gil’

In other words, A is a scalar type spectral operator [10]. Recall that the theory of the spectral operators is presented in the well-known book [10]. Let .{gk } be the set of all eigenvectors of A and .{hk } the corresponding biorthogonal sequence: .(gk , hj ) = 0, k /= j , .(gk , hk ) = 1. Then .Qk = (., hk )gk and A=

∞ 

.

λk (A)Qk .

k=1

Put T =

∞ 

.

(45)

(., hk )ek .

k=1

Simple calculations show that the inverse operator is defined by T −1 =

.

∞  (., ek )gk .

(46)

k=1

Below we check that T and .T −1 are bounded. Lemma 8 Let conditions (34)–(36) hold and T be defined by (45). Then (38) is valid with S=

∞ 

.

λk (A)Pk .

k=1

Proof Indeed, AT −1 f =

∞ ∞  

.

λk (A)(f, ej )(gj , hk )gk

k=1 j =1

=

∞ 

λk (A)(f, ek )gk (f ∈ H , T −1 f ∈ Dom(A))

k=1

and T AT −1 f =

∞ 

.

k=1

as claimed.

λk (A)

∞ ∞   (gk , hj )ej (f, ek ) = λk (A)(f, ek )ek = Sf, j =1

k=1

⨆ ⨅

Similarity Condition Numbers

123

Introduce the operator ∞ 

J =

.

‖hk ‖(., ek )ek .

k=1

Then Tf − Jf =

∞ 

.

‖hk ‖(f, hˆ k − ek )ek (f ∈ H ), where hˆ k = hk /‖hk ‖.

k=1

Hence, ‖Tf − Jf ‖2 =

∞ 

.

‖hk ‖2 |(f, hˆ k − ek )|2 ≤ ‖f ‖2

k=1

∞ 

‖hk ‖2 ‖hˆ k − ek ‖2 .

(47)

k=1

It is clear the .hk are the eigenvectors of .A∗ . Besides .‖A∗ − D ∗ ‖ = ‖A − D‖ = ς Applying Lemma 7 with .A2 = A∗ , A1 = D ∗ , according to (44) we can write ‖em − hˆ m ‖ ≤

.

2δm 2ς . = 1 − δm dm − 2ς

Now (47) implies ‖T − J ‖2 ≤ (2ς )2

∞ 

.

k=1

‖hk ‖2 . (dk − 2ς )2

(48)

We always can take .hk and .gk in such a way that ‖hk ‖ = ‖gk ‖.

.

Clearly, .Qk hk = (hk , hk )gk . So (Qk hk , gk ) = (hk , hk )(gk , gk ) = ‖hk ‖4 = ‖gk ‖4 .

.

Hence, ‖hk ‖4 ≤ ‖Qk ‖‖hk ‖‖gk ‖ = ‖Qk ‖‖hk ‖2 .

.

Thus ‖hk ‖2 ≤ ‖Qk ‖ and ‖gk ‖2 ≤ ‖Qk ‖.

.

(49)

124

M. Gil’

Now (48) implies ‖T − J ‖2 ≤ (2ς )2

∞ 

.

k=1

‖Qk ‖ . (dk − 2ς )2

Moreover, by (44), ‖Qk ‖ ≤ ‖Pk ‖ +

.

ς ς =1+ ≤ c02 (k = 1, 2, . . .), dk − ς dk − ς

(50)

where  .c0 = 1+



1/2

ς

=

dˆ − ς

dˆ

dˆ − ς

.

Consequently, ‖T − J ‖2 ≤ (2ς c0 )2

∞ 

.

k=1

1 . (dk − 2ς )2

Hence, ‖T ‖ ≤ ‖J ‖ + ‖T − J ‖ ≤ ‖J ‖ + 2c0 ς

∞ 

.

k=1

1 (dk − 2ς )2

1/2 .

But due to (49) and (50), ‖Jf ‖2 =

∞ 

.

‖hk ‖2 |(f, ek )|2 ≤ c02

k=1

∞ 

|(f, ek )|2 = ‖f ‖2 c02 (f ∈ H ).

k=1

Thus we obtain ⎛ ‖T ‖ ≤ c0 ⎝1 + 2ς

∞ 

.

k=1

1 (dk − 2ς )2

1/2 ⎞ ⎠.

The same arguments along with (46) and (49) give us the inequality ⎛ ‖T −1 ‖ ≤ c0 ⎝1 + 2ς



∞ 

.

k=1

1 (dk − 2ς )2

1/2 ⎞ ⎠,

Similarity Condition Numbers

125

and therefore, ⎛ κT = ‖T −1 ‖‖T ‖ ≤ c02 ⎝1 + 2ς

∞ 

.

k=1

1 (dk − 2ς )2

1/2 ⎞2 ⎠ = ϑ(A), ⨆ ⨅

as claimed.

12 Unboundedly Perturbed Selfadjoint Operators In the present section we estimate the condition number of a linear operator H in H with the following property: there is a positive definite selfadjoint operator S with a discrete spectrum, such that .Dom (H ) = Dom (S), and for some .ν ∈ [0, 1),

.

qν := ‖(H − S)S −ν ‖ < ∞.

(51)

.

It this section it is supposed that .σ (S) is algebraically simple. So .d1 := (λ2 (S) − λ1 (S))/2 > 0 and .dk = 21 min{λk+1 (S) − λk (S), λk (S) − λk−1 (S)} > 0 .(k ≥ 2). It is also assumed that 2qν λνm+1 (S) < dm (m = 1, 2, . . .)

.

(52)

and ϕν (S) :=

∞ 

λ2ν k+1 (S)

k=1

(dk − 2qν λνk+1 (S))2

.

< ∞.

(53)

Under condition (52) and (53) we have ην (S) := sup

.

m

dm < ∞. dm − qν λνm+1 (S)

(54)

Theorem 6 Let the spectrum of a positive definite operator S be algebraically simple. Let conditions (51)–(53) hold. Then there are a bounded and boundedly invertible operator .T , and a normal operator M, acting in .H , such that T H x = MT x (x ∈ Dom(H )).

.

Moreover,  2  κT ≤ ην (S) 1 + 2qν ϕν (S) .

.

(55)

126

M. Gil’

The proof of this theorem can be found in [23]. The theorem is sharp: if .H = S is selfadjoint, then .qν = 0 and .ην (S) = 1; we thus obtain .κT = 1.

13 Applications of Condition Numbers Let H be a scalar type spectral operator operator on .H with a discrete spectrum: Hx =

∞ 

.

λk (H )Ek x (x ∈ Dom(H )),

(56)

k=1

where .Ek , k = 1, 2, . . ., are the uniformly bounded eigen-projections of H and the eigenvalues of H are enumerated in the non-decreasing order of their absolute values with the algebraic multiplicities taken into account. Let H be similar to a normal operator D: .H x = T −1 DT x (x ∈ Dom(H )). Let .ΔPk be the eigen-projections of D. Making use of the Riesz integral representation for the eigen-projections, with small enough .r > 0 we have  2π iEk =



.

|z−λk (H )|=r

Rz (H )dz =

 .

 .

=

|z−λk |=r

(T

−1

=

|z−λk |=r

(zI −D)T )

−1

|z−λk |=r)

(zI − H )−1 dz

(zT −1 T − T −1 DT )−1 dz 

dz=

|z−λk |=r

T −1 (zI −D)−1 T dz=2π iT −1 ΔPk T

(λk = λk (H ) = λk (D)). So .Ek = T −1 ΔPk T and therefore

.

‖Ek ‖ ≤ κT (k = 1, 2, . . .).

.

(57)

Let .f (z) be a scalar function defined and bounded on the spectrum of H . Define the function of H as f (H ) =

∞ 

.

f (λk (H ))Ek .

k=1

Then we easily get f (H ) = T −1

∞ 

.

k=1

Hence we get the following result.

f (λk (H ))ΔPk T .

Similarity Condition Numbers

127

Corollary 1 Let H be defined by (56). Then .‖f (H )‖ ≤ κT supk |f (λk (H ))|. Now we can apply the above derived bounds for .κT . In particular, with .α(H ) = supk Re λk (H ) < ∞, we have ‖eH t ‖ ≤ κT (H )eα(H )t (t ≥ 0),

.

and ‖Rλ (H )‖ ≤

.

κT (λ /∈ σ (H )), ρ(H, λ)

(58)

where .ρ(H, λ) = infk |λ − λk (H )|. Let A and .A˜ be linear operators. Then the quantity ˜ := sup svA (A)

.

˜ t∈σ (A)

inf |t − s|

s∈σ (A)

is said to be the spectral variation of .A˜ with respect to A. Now let .H˜ be a linear operator on .H with .Dom(H ) = Dom(H˜ ) and q := ‖H − H˜ ‖ < ∞.

.

(59)

From (58) it follows that .λ /∈ σ (H˜ ), provided .qκT < ρ(H, λ). So for any .μ ∈ σ (H˜ ) we have .qκT ≥ ρ(H, μ). This inequality implies our next result. Corollary 2 Let H be defined by (56) and condition (59) hold. Then .svH (H˜ ) ≤ qκT . Now consider unbounded perturbations. To this end put H −ν =

∞ 

.

ν λ−ν k (H )Ek , H =

k=1

∞ 

λνk (H )Ek (0 < ν < 1),

k=1

assuming that .supk |λ−ν k (H )| < ∞. We have H Rλ (H ) =

.

ν

∞ 

λνk (H )(λk (H ) − λ)−1 Ek (λ /∈ σ (H )).

k=1

Due to Corollary 1, ‖H ν Rλ (H )‖ ≤

.

κT (λ /∈ σ (H )), ψν (H, λ)

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where ψν (H, λ) := inf |(λ − λk (H ))λ−ν k (H )|,

.

k

assuming that .ψν (H, λ) > 0 for regular .λ. Now let .H˜ be a linear operator on .H with .Dom(H ) = Dom(H˜ ) and qν := ‖(H − H˜ )H −ν ‖ < ∞ (0 < ν < 1).

.

(60)

We have Rλ (H ) − Rλ (H˜ ) = Rλ (H )(H˜ − H )Rλ (H˜ ) = Rλ (H˜ )(H˜ − H )H −ν H ν Rλ (H ).

.

Thus, .λ /∈ σ (H˜ ), provided the conditions (60) and .qν κT (H ) < ψν (H, λ) hold. So for any .μ ∈ σ (H˜ ) we have qν κT ≥ ψ(H, μ).

.

(61)

The quantity ν − rsvH (H˜ ) := sup

.

t∈σ (H˜ )

inf |(t − s)s −ν |

s∈σ (H )

is said to be the .ν− relative spectral variation of operator .H˜ with respect to H . Now (61) implies. Corollary 3 Let H be defined by (56). Let condition (60) hold. Then .ν−rsvH (H˜ ) ≤ qν κT . About the recent results on perturbations of operators and operator functions see [1, 32, 35].

14 Bibliographical Comments Sections 2–5 are based on the paper [18]. The material of Sects. 6–8 is taken from [20]. The material of Sects. 9–11 is adopted from the papers [13] and [19]. As it was above mentioned, the proof of Theorem 6 can be found in [23]. That proof is particularly based on the main result from [21]. For more relevant results, which are not included into this paper see [24]. Bounds for the condition numbers of finite dimensional operators have been established in [15–17] (see also [22]).

Similarity Condition Numbers

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References 1. D. Andrica, Th.M. Rassias (eds.), Differential and Integral Inequalities, Springer Optimization and Its Applications, vol. 151 (Springer Nature, Switzerland, 2019) 2. N.-E. Benamara, N.K. Nikolskii, Resolvent tests for similarity to a normal operator. Proc. Lond. Math. Soc. 78, 585–626 (1999) 3. T. Betcke, S.N. Chandler-Wilde, I.G. Graham, S. Langdon, M. Lindner, Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation. Numer. Methods Partial Differ. Equ. 27, 31–69 (2011) 4. R. Bhatia, P. Rosenthal, How and why to solve the operator equation AX − XB = Y . Bull. Lond. Math. Soc. 29, 1–21 (1997) 5. G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems. J. Oper. Theory 53(1), 49–89 (2005) 6. G. Cassier, D. Timotin, Power boundedness and similarity to contractions for some perturbations of isometries. J. Math. Anal. Appl. 293, 160–180 (2004) 7. S.N. Chandler-Wilde, I.G. Graham, S. Langdon, M. Lindner, Condition number estimates for combined potential boundary integral operators in acoustic scattering. J. Int. Eqn. Appl. 21, 229–279 (2009) 8. G. Chen, Y. Wei, Y. Xue, The generalized condition numbers of bounded linear operators in Banach spaces. J. Aust. Math. Soc. 76, 281–290 (2004) 9. Yu.L. Daleckii, M.G. Krein, Stability of Solutions of Differential Equations in Banach Space (American Mathematical Society, Providence, 1971) 10. N. Dunford, J.T. Schwartz, Linear Operators, Part 3, Spectral Operators (Wiley-Interscience Publishers, New York, 1971) 11. M.M. Faddeev, R.G. Shterenberg, On similarity of differential operators to a selfadjoint one. Math. Notes 72, 292–303 (2002) 12. M.M. Faddeev, R.G. Shterenberg, On the similarity of some singular differential operators to selfadjoint operators. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270, Issled. po Linejn. Oper. i Teor. Funkts. 28, 336–349, 370–371; translation in J. Math. Sci. (N.Y.) 115(2) (2003), 2279–2286 (2000) 13. M.I. Gil’, Perturbations of simple eigenvectors of linear operators. Manuscr. Math. 100, 213– 219 (1999) 14. M.I. Gil’, Operator Functions and Localization of Spectra. Lecture Notes in Mathematics, vol. 1830 (Springer, Berlin, 2003) 15. M.I. Gil’, Perturbations of functions of diagonalizable matrices. Electron. J. Linear Algebra 20, 303–313 (2010) 16. M.I. Gil’, Matrix equations with diagonalizable coefficients. Gulf J. Math. 1, 98–104 (2013) 17. M.I. Gil’, A bound for condition numbers of matrices. Electron. J. Linear Algebra 27, 162–171 (2014) 18. M.I. Gil’, A bound for similarity condition numbers of unbounded operators with Hilbert– Schmidt Hermitian components. J. Aust. Math. Soc. 97, 331–342 (2014) 19. M.I. Gil’, On condition numbers of spectral operators in a Hilbert space. Anal. Math. Phys. 5, 363–372 (2015) 20. M.I. Gil’, An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components. Filomat 30(13), 3415–3425 (2016) 21. M.I. Gil’, Rotations of eigenvectors under unbounded perturbations. J. Spectral Theory 7(1), 191–199 (2017) 22. M.I. Gil’, Operator Functions and Operator Equations (World Scientific, Hackensack, 2018) 23. M.I. Gil’, On similarity of unbounded perturbations of selfadjoint operators. Methods Funct. Anal. Topol. 24(1), 27–33 (2018) 24. M.I. Gil’, Similarity of operators on tensor products of spaces and matrix differential operators. J. Aust. Math. Soc. 106(1), 19–30 (2019)

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25. M.I. Gil’, On location of the spectrum of an operator with a Hilbert-Schmidt resolvent in the left half-plane. Methods Funct. Anal. Topol. 27(4), 340–347 (2021) 26. I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, vol. 18 (American Mathematical Society, Providence, 1969) 27. I.M. Karabash, A.S. Kostenko, M.M. Malamud, The similarity problem for J-nonnegative Sturm–Liouville operators. J. Differ. Equ. 246, 964–997 (2009) 28. A. Kostenko, The similarity problem for indefinite Sturm–Liouville operators with periodic coefficients. Oper. Matrices 5(4), 707–722 (2011) 29. A. Kostenko, The similarity problem for indefinite Sturm–Liouville operators and the help inequality. Adv. Math. 246, 368–413 (2013) 30. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966) 31. M.M. Malamud, Similarity of a triangular operator to a diagonal operator. J. Math. Sci. 115(2), 2199–2222 (2003) 32. P.M. Pardalos, T.M. Rassias (eds.), Mathematics Without Boundaries, Surveys in Interdisciplinary Research, vol. VIII (Springer, New York, 2014) 33. S.V. Parter, S.-P. Wong, Preconditioning second-order elliptic operators: condition numbers and the distribution of the singular values. J. Sci. Comput. 6(2), 129–157 (1991) 34. B. Pruvost, Analytic equivalence and similarity of operators. Integr. Equ. Oper. Theory 44, 480–493 (2002) 35. Th.M. Rassias, V.A. Zagrebnov (eds.), Analysis and Operator Theory. Dedicated in Memory of Tosio Kato’s 100th Birthday. Foreword by Barry Simon (Springer, Berlin, 2019) 36. M. Rosenblum, On the operator equation BX − XA = Q. Duke Math. J. 23, 263–270 (1956) 37. M. Seidel, B. Silbermann, Finite sections of band-dominated operators, - norms, condition numbers and pseudospectra, in Operator Theory: Advances and Applications, vol. 228 (Springer, Basel, 2013), pp. 375–390

Legendre’s Geometry and Trigonometry at the Evelpides School (Central Military School) During the Kapodristrian Period Andreas Kastanis

1 Introduction The development of the sciences is closely related to the state of education. Education, in turn, is linked to the existence of educational and scientific books. One of the most interesting branches of science is mathematics. But what were the mathematics books during the Kapodistrian period 1828–1832 and what position did Legandre’s books have? What use were the translations of Legendre’s books at the Central Military School? The answer to these questions will allow us to trace, both the state of mathematics and the position that Legendre’s Geometry and Trigonometry had, during the above period of time.

2 Mathematical Education in General The prevalence of Enlightenment ideas led the Greeks to their national restoration with the National Revolution of 1821. In this context, educational policy, in general, served two purposes: the Europeanization of Greek education and the shaping of the national identity of the Greek people. After the descent of Kapodistrias in Greece, in January 1828, education was organized with the first goal of developing elementary education. The schools of Aegina [28] were organized, as well as the precursor of the Military School of Evelpida, the Central Military School or Central Military School. The two schools Central and Central Military were organized according to French standards, the

A. Kastanis () Greek Military Academy, Vari Attikis, Greece © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_8

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Central according to the standards of the Ecole Centrales, and the Central Military according to the standards of the Ecole Polytechnique [26]. Kapodistrias was well aware of the need for education and the lack of suitable textbooks for the schools of the territory. By its resolution, committees are set up with the main task of writing, translating, or revising the already translated books necessary for the educational needs of the newly established Greek state [28, p. 22]. The appointment of the French army captain Henri-Auguste Dutrône [French Philhellenic. Adviser to the Greek Government on educational issues] as a member of one of the above committees [28, p. 23] shows the strong presence of French ideas in the selection of educational books. In the wider Greek educational effort, the Ionian Academy was a pioneer in the teaching of physics and mathematics. The main contributor to this event was her professor Ioannis Karantinos, who integrated the spirit of French mathematics into her program.

3 Ioannis Karantinos The Ionian Academy played a prime role in the teaching of physics and mathematics. The main contributor to this was Ioannis Karantinos. Born in Kefalonia in 1784, at the age of 20 Karantinos was enrolled in the army of the Ionian Islands. In 1808, he entered the Academy of Corfu, having the French Charles Dupin (1784–1873) captain of the Engineering corps, as his teacher of sciences. Later he studied at the École Polytechnique. In 1824, Karantinos was appointed professor of mathematics at the Ionian Academy. He translated and published a number of books on mathematics. Thanks not only to his editorial work but also to his students, Karantinos played an important role in the development of mathematics in Greece. He died insane in 1834. Karantinos incorporated the spirit of French mathematics in the curriculum. For the needs of teaching, he translated Bourdon (1828), Leslie (1829), Legendre (1828 & 1830). These books contributed to establish mathematical education in Greece; they were used as teaching manuals in many schools [Belia 1970, 87]. In addition, Karantinos translated the following books, without publishing them: almost one half of Lagrange’s (1797), Biot (1823), Bourdon (1823), Poisson (1809), Santini (1819–1820). It is worth noting here that the governor Kapodistrias himself asked Karantinos to translate Bourdon (1821) in order to meet the needs of the Greek schools. For the needs of teaching, he translated and published the following books: α. Bourdon, L.P.M., 1821. Éléments d’arithmétique. Courcier, Paris. Transl. I. Karantinos: ∑τoιχε´ια Aριθμητικης ´ . Anton Haykul,Vienna, 1828. β. Legendre, A.M., 1823. Éléments de Géométrie, 12th Edition. Didot, Paris. Transl. I. Karantinos: ∑τoιχε´ια 𝚪εωμετρ´ιας Vol. 2, 1st Edition. Government Typography, Corfu, 1828.Στ oιχ ε´ια Γ εωμετ ρ´ιας τoυ Legendre, τo 1828

Legendre’s Geometry and Trigonometry at the Evelpides School (Central. . .

133

γ. Leslie, J., 1829. Aναλυσις ´ 𝚪εωμετρικη´ Iωαννη ´ Λεσλ´ιoυ (transl. I. Karantinos, Vol. 3). Government Typography, Corfu. δ. Legendre, A.M., 1823. Éléments de Géométrie,1 12th Edition. Didot, Paris. Transl. I. Karantinos: I. Karantinos: ∑τoιχε´ια Tριγωνoμετρ´ιας [14, p. 54]. The above books, within the framework of the general educational policy, shaped mathematical education in Greece and were used, at least during the Capodistrian period, as educational aids in many schools [24, p. 87] With these thoughts, Kapodistrias approved [25, Vol C, p. 102] the request of Ioannis Karantinos, who in a letter asked him to be reinforced with the amount of 600 Spanish thalers,2 in order to publish Legendre’s Geometry and Trigonometry [1]. The amount would be given in three installments with the obligation to send to Greece a number of copies of Bourdon’s3 Arithmetic and Algebra. If there was money left over, he asked for copies of Legendre’s Geometry [25, vol C, p 102]. As a result of this funding, Karantinos, in September 1829, sent out 200 copies of Bourdon’s Arithmetic and Analytical Geometry which was published before Legendre’s [27, p. 98 and 3]4 Geometry [4]. The following year, in February 1830, 200 books of Legendre’s Geometry and around the end of the same year 200 copies of Trigonometry [8] were received by Moustoxidis.5 It should be taken into account that for the first time an independent Trigonometry book appears in the Greek mathematical literature [17]. According to a balance sheet of the Secretariat of Ecclesiastical and Public Education, 600 books of the Geometry of the Legend, translated by I. Karantinos, were bought and 9000 Phoenixes were given from February 1828 to the end of August 1830 [12]. The amount of 9000 phoenixes was the largest given for the purchase of books.6 A total of 800 copies of Geometry were purchased, i.e., 200 initially (the initial shipments) and an additional 600. Of the 600 copies, up to July 1 Legendre’s

French book of Geometry also contained Trigonometry until 1845. thalers (one Spanish thaler corresponded to 6 Phoenixes (Greek coin)) 3 Bourdon Louis Pierre Marie was born in 1799. He studied at the Ecole Polytechnique in France in 1796. In 1804 he was a high school mathematics teacher and examiner at the Ecole Polytechnique. He wrote Elements of Algebra in 1817, Elements of Arithmetic in 1821, and Applications of Algebra to Geometry and Trigonometry. His books were the exam material for candidate students at the Ecole Polytechnique 4 Andrian-Marie Legendre. In 1775 he became a professor at the Military School of Paris. In 1815 he was an examiner at the Ecole Polytechnique. He became known mainly with his book Elements of Geometry (including Trigonometry). This Geometry was the main cause of Euclid’s abandonment. He wrote many other studies in mathematics. 5 Andreas Moustoxidis’ (1785–1860) (director of the Central School of Aegina and the Orphanage of Aegina) 6 Other books bought were from Korais and the amount given to him was a total of 6590 phoenixes which were the first Greek money. 2 Spanish

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15, 1832, 520 were distributed to the schools and the rest remained in the National Library of Aegina 28 and the warehouses of the Orphanage7 28 [9, 12]. Along with the financing of the Karantinos’ editions, teacher Pampoukis8 also submitted a similar request to the Governor. He requested the help of Kapodistrias to translate and publish Legendre’s geometry and trigonometry [2]. Kapodistrias shows a special interest and asks on the same day to be informed about the cost of this edition.9 There is no information about the fate of this translation by Pampoukis.10 [15, p. 679] According to the National Gazette, which published the program of the Central School, the mathematics taught was Legendre’s Geometry and Trigonometry translated by Karantinos.

4 Ecole Centrale Militaire – Central Military School (Kεντρικ´o Πoλεμικ´o ∑χoλε´ιo) One of the main concerns of Ioannis Kapodistrias (1796–1831), the first governor of independent Greece, was to organize the army and the education. The foundation of the Greek military Academy on July 1, 1828 is in this context. He was tried to create a national army obedient to the commands of the government and in addition architects who were to undertake the reconstruction of Greece. The Military Academy was called “Company of the Evelpides” (Λ´oχoς των Eυελπ´ιδων). The only appointed professor we know of was the mathematician Dimitrios Despotopoulos a student of Ioannis Karantinos (1784–1834). On December 2nd 1828, Jean-Henri-Pierre-Augustin Pauzié (1792–1848), a former student of École Polytechnique who served as a captain of the French Artillery Corps suggested to Kapodistrias, the establishment of a Military Polytechnic School (Πoλυτεχνικ´o ∑τρατιωτικ´o ∑χoλε´ιo) [25, vol B, p. 322–323]. The governor was sympathetic to this proposal, and entrusted Pauzié to suggest the organizational plan of the new School. Colonel was ordered to co-operate with Pauzié in this effort [25, vol B, p. 338]. Both men signed the plan on December 28th 1828. It was written in French and divided into seven chapters to amount 109 articles [13].11 7 The

orphans of the war were one of Kapodistrias’ main concerns. He ordered an orphanage to be built in Aegina, which came into operation by 1829. This orphanage provided primary education as well as vocational training for some technical jobs such as tailors, goldsmiths, blacksmiths, etc. A few orphans joined the Central Military School afterwards. Moreover, as the orphanage was a spacious building, it was also used as a book storage, which later became the first National Library of Greece. 8 Pampoukis was initially a teacher at the Greek school of Aegina and later a teacher at the school of Akrata and Arachova. 9 Letters of I.A. Kapodistria as above part C, p 102 10 According to information, Koletis had translated Legendre’s Geometry and Biot’s arithmetic, but they were never published. 11 The titles of the chapters were: A. general regulations, B. several regulations with the following divisions: Military organization about arming and equipment, clothing, billeting and military

Legendre’s Geometry and Trigonometry at the Evelpides School (Central. . .

135

The École Polytechnique appears to have played an effective role in the French cultural penetration in Greece. It also served as a model for the dominant role of mathematics in the teaching curriculum [16, 18]. First year’s geometry “the first four chapters of Legendre (1823)”. Second year’s mathematics included “chapters 5, 6 and 8 of Legendre (1823)”, trigonometry including the use of sine tables.

4.1 Geometry When Pauzié and Heideck12 undertook the responsibility to organize the Academy (Central Military School), they decided (organizational plan of 1829) to distribute as a teaching handbook Legendre (1823). This choice was an easy one for the organizers of the Central Military School: Legendre (1823) was one of the few French handbooks of geometry, which had been translated into Greek [21]. Legendre (1828) was the main handbook of mathematics in Greece during the nineteenth century. With this book Legendre was considered as the second Euclid. It should be noted that despite all the similarities between the two Geometries there are differences, the main one of which was the use of Arithmetic and Algebra in the field of Geometry [27, p. 7]. It was first published in 1794 in Paris under the title Eléments de Géométrie [29, p. 487]. They included eight books (chapters), of which the first four chapters referred to planimetry and the next four to stereometry. Specifically, the contents (Fig. 1) were: The broad acceptance of this book is confirmed by its many translations into Greek [27, p. 9]. Apparently, Ioannis Koletis (1774–1847)13 and Georgios Kalaras14 had translated this book; however, it was not printed and no manuscript has been found. Then Kapodistrias encouraged Charalabos Paboukis15 [28, p. 126] to

uniforms, C. courses (articles 67–83), D. wages and responsibilities, E. discipline and penalties, F. special regulations and G. special courses 12 Karl von Heideck (1788–1861). Bavarian general. He studied in Munich and served in the Bavarian army. As a lieutenant, he fought on the side of Austria and later, in 1813, in the French army against Spain. He came to Greece during the Greek Revolution. He participated in many battles. In 1828 he was appointed by Kapodistrias governor of Nafplion. In August 1829 he returned to Bavaria but was reinstated as a member of Otto’s Regency. He remained in Greece until 1835. He wrote memoirs about the Greek Revolution, which were published in the journal Armonia. 13 Koletis was one of the leaders of the Greek Revolution and in 1844 was elected Prime Minister of Greece. 14 It is unknown when he was born and died 15 Paboukis was first teacher at the Greek school of Aegina, later he taught at the schools of Akratas and Arachovas. It is unknown when he was born and died

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Title

I

Fundamental Principals

II

The circle

III

Proportions

IV

Regular polygons

V

Plains

VI

Polyedrons

VII

The sphere

VIII

The three round bodies [23 p. 1, 25, 45, 85, 11, 131, 162, 199]

Fig. 1 Table of contents

translate this book providing him with all the necessary assistance [25, vol. C, p. 101 and 2]. Kapodistrias also asked Karantinos for the translation of this book (Fig. 2) in order to meet the needs of the schools [25, vol. C, p. 102]. According to both the content of Legendre’s book and the article 50 of the 1829 organizational plan, the subjects to be taught were shared out in such a way that the lowest grade was taught plane geometry while the second-grade geometry of solids, except chapter “Bounds for Similarity Condition Numbers of Unbounded Operators” which was about the sphere. In order to cover the needs of the Central Military School, there were given 40 copies [4, 10], so that each Cadet could have his own manual and was not obliged to keep notes during the course. The new organizational plan of 1834 [5] continued the teaching of Legandre’s Geometry. The academic Council of the Academy decided that Geometry should be taught only in the two preliminary ranks. Initially, the first four chapters (books) of Legendre’s Geometry (Planimetry) would be taught in the 3rd department of D rank and in the 2nd department of D rank the teaching of the last four books of the same writer (Stereometry) [5] would be continued. The choice of the Legendre Geometry [26, p. 83 and 27, p. 537–538].16

16 Educational

period

handbook and the subjects taught were the same with those of the Kapodistrian

Legendre’s Geometry and Trigonometry at the Evelpides School (Central. . .

137

Fig. 2 The first Greek edition of Legendre’s Geometry translated by Karantinos

The Legendre Geometry dominated the Greek mathematical education during the entire nineteenth century. It was translated many times by various writers. It was used as an instructive handbook in Greek high schools [5], but it was also included in 1837 in the University curriculum and taught there.

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4.2 Trigonometry Although there is no relevant information about trigonometry in the organizational plan of 1829, there is a request submitted by the Central Military School to the Orphanage of Aegina for 20 copies of Legendre (1830) [6, 8]. The reasons for the choice of this particular handbook were the same as for the rest of the mathematical handbooks. Even though there were books such as Kouma (1807), whose third volume concern trigonometry [20, p. 57], Pauzié and Heideck did not use it. Apparently, it did not represent the French model that they wanted to apply to the Central Military School. The government had ordered Karantinos, apart from all the other books, to produce copies of Legendre’s Trigonometry [7]. No manual was planned to be given to the Cadets for the lectures of this course. Karantinos’ translation [19] is 183 pages long (Fig. 3). The first 40 pages are an introduction, pages 40 to 100 contain the treatise on trigonometry proper while pages 101 to 117 constituted the appendix, which including answers to ‘several questions on trigonometry’. Finally, pages 119 to 183 include notes related to Legendre (1823). There is no table of contents. This course was taught in the second grade. Trigonometry continued to be taught in the 4th preparatory grade [23], even after the reorganization of the Academy in 1834.

5 Conclusion The prevailing opinion, at this period, in France was that mathematics constituted the most necessary science of all others for the officers of the army, especially the Artillery and Engineers. Since the eve of the French Revolution, there has been a rapid growth in the teaching of this science in military schools. The interest arose not only from the practical application, which they had in Artillery and Engineering but also in the Infantry, because the teaching of mathematics would help to develop the ability to reason thought, an element absolutely necessary for all soldiers, regardless of Arm. After all this, mathematical thought began to be more and more connected with military thought. Initially in 1785 at the imperial military school in Paris only 1/8 of the program was devoted to Mathematics. Later, at the Ecole Polytechnique of France, the positive sciences occupied a very large part of the content of the studies. The program of teaching Mathematics had similarities with the corresponding program of the Ionian Academy, which in turn, at least in, was influenced by the Ecole Polytechnique of France. This effect was due to the fact that the professor of this course, Ioannis Karantinos, studied in 1820 at the Ecole Polytechnique.

Legendre’s Geometry and Trigonometry at the Evelpides School (Central. . .

139

Fig. 3 The first Greek edition of Legendre’s Trigonometry translated by Karantinos

In conclusion, there were three main factors that helped for this choice. The first was the function of the Ionian Academy in which Ioannis Karantinos translates and teaches Legendre’s Geometry. When his students came to Greece, they spread French mathematics and, by extension, Geometry. The second was the translation and printing of a series of French mathematics textbooks ordered and purchased by the Greek Government. The third was the organization of the Central Military School by the French who imposed the teaching of Legendre’s geometry and French mathematics in general.

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References General National Archives of the Greek State (GAK) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17.

18. 19. 20. 21.

22.

23. 24.

25.

GAK General Secretariat 19 April 1829 f 196 GAK General Secretariat 2 May 1829 f 198 GAK General Secretariat 24 July 1829 f 211 GAK General Secretariat, 2 May 1830 f 5 GAK Organization of the Evelpidon Military School, 17 March 1842 f 372 GAK Secretariat of Ecclesiastical and Public Education 26 November 1830 f 33 GAK Secretariat of Ecclesiastical and Public Education 8 July 1830 f 30 GAK Secretariat of Ecclesiastical and Public Education 9 December 1830 f34 GAK Secretariat of Ecclesiastical and Public Education, 18 July 1832 f 51 GAK Secretariat of Ecclesiastical and Public Education, 20 September 1830 f 31b GAK Secretariat of Ecclesiastical and Public Education, August 1830 f 30b GAK Secretariat of Ecclesiastical and Public Education, July 1832, Appendix B of document, f 51 GAK Vlachoyannis collection: Department of Army & Navy 21 February 1829, f 102 Final report of the National Committee of fifteen on Geometry syllabus. Math. Teach. 5, (1912–13) (Π𝚪AK) Tα Περιεχ´oμενα των 𝚪ενικων ´ Aρχε´ιων τoυ Kρατoυς, ´ τ 5oς αρ 15α εκδ 𝚪AK Aθηνα ´ 1976 (The Contents of the General Archives of the State t 5th no 15th ed GAK Athens 1976) B. Belhoste, A. Dahan- Dalmédico, A. Picon (eds.), La Formation Polytechnicienne, 1794– 1994 (Dumond, Paris, 1994) Kastanis A, (Kαστανης ´ Aνδρϵ´ ας), 1997. « To μαθηματικ´o βιβλ´ιo κατα´ την περ´ιoδo 1828–1932», Aϕιϵ´ ρωμα στoν Aντωνη ´ Aντωνακ´oπoυλo, επιμ K. Aρωνη´ Tσ´ιχλη, Παπαζηση ´ (Kastanis Andreas “The mathematical book during the period 1828–1932”, Tribute to Antonis Antonakopoulos) J. Langnis, 1987. La République avait besoin de savants. Paris: Belin 1990, The École Polytechnique and the French Revolution: Merit, Militarization and Mathematics, LLUL 13 Legendre, ∑τoιχε´ια Tριγωνoμετρ´ιας translated from French to Modern Greek dialect by Dr. Ioannis Karantinos from Kefallinia 1830, vol 6 Corfu: Government Typography A. Poulos, Eλληνικ η´ Mαθηματ ικ η´ Bιβλιoγ ραϕ´ια (1500–1900) (Eλληνικη´ Mαθηματικη´ Eταιρε´ια (Hellenic Mathematical Association), Athens, 1988), p. 57 K. Xanthopoulos, 1880, ∑υνoπτικη´ ´Eκθεσις της Πνευματικης ´ Aναπτυξεως ´ των Nεωτϵ´ ρων Xρ´oνων . . . αναγεννησεως ´ αυτων ´ μϵ´ χρι τoυδε. ´ Kωσταντινoυπoλη ´ (Summary Report of the Spiritual Development of the Newer Times... rebirth of them up to now. Constantinople): Boυτυρας ´ (reprind 1988 Athens: Kαραβ´ια) A.M. Λεγϵ´ νδρoυ, ∑τoιχε´ιαs 𝚪εωμετρ´ιας μεταϕραση ´ Iωαννη ´ Kαραντινoυ, ´ εκδ δευτερη, ´ 1840, (Legendre, A.M. 1823 Eléments de Géométrie 12 ed. Paris : Chez Firmin Didot, Père et fils (∑τoιχε´ια 𝚪εωμετρ´ιας, translated by Ioannis Karantinos (1828, 1 ed), (1840, 2 ed) Vol 2 Corfu : Government Typography) EK, Eϕημερ´ις της Kυβερνησεως, ´ 17 August 1834 (Official Government Gazette) Eλϵ´ νη Mπελια, ´ H Eκπα´ιδευση στη Λακων´ια και την Mεσσην´ια κατα´ την Kαπoδιστριακη´ Περ´ιoδo 1828–1832, Aθηνα, ´ 1970, σ 87. (Eleni Belia, Education in Laconia and Messinia during the Kapodistrian Period 1828–1832, Athens, 1970, p 87) Eπιστoλα´ι I.A.Kαπoδ´ιστρια Kυβερνητoυ ´ της Eλλαδoς. ´ Διπλωματικα´ι, Διoικητικα´ι και Iδιωτικα´ι, γραϕε´ισαι απ´o 8 Aπριλ´ιoυ 1827 μϵ´ χρις 26 ∑επτεμβρ´ιoυ 1831. Eκδoθε´ισαι παρα´ E.A Bεταν ´ τ´oμoς 𝚪 Aθηνα ´ 1842 (Letters of I.A. Kapodistria, Governor of Greece. Diplomatic, Administrative and Private, written from April 8, 1827 to September 26, 1831. Published by E.A. Betan volume C Athens 1842)

Legendre’s Geometry and Trigonometry at the Evelpides School (Central. . .

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26. Kαστανης ´ Aνδρϵ´ ας H ∑τρατιωτικη´ ∑χoλη´ των Eυελπ´ιδων κατα´ τα πρωτα ´ χρ´oνια της λειτoυργ´ιας της 1828–1834 διδακτoρικη´ διατριβη, ´ Iωαννινα ´ 1995 (Kastanis Andreas The Evelpid Military School during the first years of its operation 1828– 1834. PhD, Ioannina 1995) 27. Kαστανης ´ N « H Eπ´ιδραση των 𝚪αλλικων ´ μαθηματικων ´ στην Nεoελληνικη´ Παιδε´ια την περ´ιoδo 1800–1840» εκδ ´Oμιλoς για την Iστoρ´ια των μαθηματικων. ´ Eλληνικη´ εταιρε´ια ιστoρ´ιας επιστημων ´ και τεχνoλoγ´ιας. Mαρτιoς ´ 1994 (N. Kastanis “The Influence of French mathematics on Modern Greek Education in the period 1800–1840“ published by the Group for the History of Mathematics. Greek Society of History of Science and Technology) 28. Koυκκoυ ´ Eλϵ´ νη O Kαπoδ´ιστριας και η Παιδε´ια (1827–1832) B’ Tα εκπαιδευτικα´ Iδρυματα ´ της Aιγ´ινης εκδ τρ´ιτη Aθηνα ´ 1989 (Eleni Kukkou, The Kapodistrias and Education (1827–1832) B’ The educational institutions of Aegina third edition Athens 1989) 29. D.E. Smith, History of Mathematics, vol 1 (Dover, New York, 1958)

The Overshadowing of Euclid’s Geometry by Legendre’s Géométrie in the Modern Greek Education Nikos Kastanis

1 Motivation In a historical study about the history of school geometry in Modern Greek education, it was noted that In Greece up to the end of the 19th century, the geometry school textbook that prevailed in secondary education was Legendre’s geometry in several versions. But even during the following period up to 1968, many Legendre-type geometry books were published . . . (see [44], p. 493).

Moreover, on comparing the European mathematical culture of the nineteenth century, and in particular of Italy, with Greece it was pointed out that In Greek mathematics education, after 1832, “can be observed a competition between German and French influences. However one might have expected nationalism here too, extolling Euclid as the native Greek “national” tradition. This was not the case, however: textbooks, as prescribed and used in the Hellenic School (lower form) and the Gymnasium (upper form) since 1840 were largely translations or adaptations either German or French ones. With regard to the French books, there was no criticism of Legendre: rather he was used over a long period, and several different translations or adaptations were published during the nineteenth century” (see [34], p. 384).

But what did really happen in Greece with Legendre’s Géométrie? Was its impact that magnificent? Furthermore, was there such a national negligence by the Greeks in the nineteenth century, about Euclid?

N. Kastanis () Department of Mathematics, Aristotle University of Thessaloniki, Greece © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_9

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2 The First Clues of the Greek Mathematics Context In 1806, Georgios Kalaras (ca. 1781–1825), who studied medicine and mathematics at the University of Pisa, announced the translation of three French mathematics books in Greek language, among them was the Éléments de Géométrie by A.M. Legendre. A few years later, 1812, Ioannis Kolettis (1773/4–1847) elaborated another Greek translation of the Legendre’s Géométrie, ready for publication. It is worth noticing that he became Prime Minister of Greece for the period 1844–1847 (see [20], p. 178, 200). He studied medicine, such as Kalaras, in Pisa during the first decade of the nineteenth century. Both of these translations were never published, and the corresponding manuscripts were not found. This is an indicating fact that they had no influence on the Greek culture at that time. However, this historical information created some questions; what did motivate them to that action? What did suspend their initiative? And finally, which were the available textbooks of geometry in Greek language, in that time? The first clue that emerges from the background of the prospective Greek translators of the geometry of Legendre is the mutual context of their studies; Medical study in the University of Pisa, during the first decade of the nineteenth century. Thus, a memorable line of the impulse towards this Geometry, in the education of Pisa at the time, must have been the influence of the French postrevolutionary ideas and works. Furthermore, there must have been some relevance of medical studies with mathematics and geometry in particular. Pisa had indeed received strong French influence in the early nineteenth century and medical studies were connected with the mathematics education, where Legendre’s Géométrie played a central role. That situation, did not only affect politically and ideologically a big part of Italians and foreigners that resided there, but there was also a relative impact on various social issues, such as the structure and content of university courses. A feature of medical studies in northern Italy at that time was the support with a physics-mathematics background.1 Notably, in 1803, in the curriculum of the first year of medical studies there was the course: Principles of Geometry and Algebra (see [2], p. 111). If you even take into account that Legendre’s Géométrie was translated and published for the first time in Pisa, in 1802, it is clear that this textbook had a greater impact on medical studies there, during this period of strong French influence in northern Italy (see [35], p. 262). So, the corresponding enthusiasm of the two Greek students, of the Medical Faculty of the University of Pisa, was really natural. However, we don’t know the level of the Greek education that the two translators of Legendre’s Géométrie had before being registered in the Medical School of Pisa. Certainly they were not ignorant when they left to study in Italy. However, there

1 Traditionally, at some universities of Northern Italy, there had been a Medical-Physics Faculty, composed of two Departments: a preparatory one, for Physics-Mathematics and a Medical-Surgical (see [36], p. 97).

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is a touch of the Greek educational contexts of their origin places that defuses in their pre-university educational opportunities. Georgios Kalaras was born and raised, during 1800, in the region of Corinth in the Peloponnese, where there wasn’t any notable school (see [8], p. 334). The most important Greek school in the region was that of Dimitsana. Therefore, his pre-university geometric knowledge would, at best, be equivalent to the respective school classes of Dimitsana. Ioannis Kolettis was born and raised in a village in the region of Ioannina, in Epirus, which was one of the major centers of Greek culture. These indications of cultural contexts of the two Greeks, who attempted the translation and publication of Legendre’s Géométrie in Greek, indicate some favorable conditions of their geometric pre-university knowledge. That is quite likely not to have been geometrically “ignorant” when they went to Pisa. However, their geometrical knowledge, from their circular training, was ideologically neutral or, at worst, where surrounded by an indirect negativity. That’s because, until the late eighteenth century, the Greek national conscience, which descended from the ancient Greek heritage and put forward as a special collective ideal, did not exist (see [13], p. 83, 113, [47], p.146–147, [48], p. 51–53). In other words, Euclid and his monumental work, The Elements, did not constitute part of their cultural identity. Generally, that is because the spirituality and attitude of the Orthodox Church did not favor the development of an ancient-Greek collective conscience (see [18], p. 521). Along with the indifference of the establishment of the Orthodox Church with regard to Ancient Greek heritage and Euclid, in particular, foster a silent cautiousness or detachment from the works and studies of the “Propaganda fide”, which systematically and methodical cultivated it’s model to the culture of the Orthodox communities (see [21], p. 60). The Euclid’s Elements was one of the “tricky channels” of this intrusion. This educational promotion has had good results by introducing Euclid as a reputable course in education, provided by the Jesuits and other Roman Catholic monks, acting as missionaries among the Orthodox Greeks under Ottoman rule (see [21], p. 56), or playing an important role in introducing Greek translations of Latin books, for example, that of Euclidean Geometry by Jesuit editions or even its variations, written by well-known clergymen of the Catholic Church (see [24], p. 93).It is worth mentioning, that the publication of the Greek translation of Euclid’s Elements, which was adapted to the spirit of seventeenth century by Jesuit André Tacquet, took place in 1805. It is one of the two autonomous, specialized Geometry textbooks, which existed in Greek education up the early nineteenth century. Therefore, it is noticed that when it was attempted to translate Géométrie of Legendre into Greek, in the early nineteenth century, there existed a satisfactory amount of manuals of Synthetic geometry in Greek mathematics education. There were two kinds: according to the model of Euclid2 or undocked from this

2 To this category belong the books-translations of Anthracites, Theotokis, Voulgaris and Benjamin of Lesvos (see [39]).

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model.3 In that time, powerful trends of modernization and secularization of Greek culture and Greek mathematics education in particular, were developed. And in that context the introduction of the Géométrie of Legendre was rather consistent and welcome. However, the conservative establishment of the Orthodox Church, which had the intellectual and institutional control of the Greek culture, not only did not favor such modernizations, but persecuted them, ideologically and administratively (see [40], p. 59). That’s why Kalaras’s and Kolettis’s initiatives for translations did not thrive.

3 Change of Context, New Potential In 1829 was published, for the first time, a Greek translation of Ioannis Carandinos (1784–1834) of the Géométrie of Legendre. The motivation for this initiative was the instructional needs of the Central Military School, founded in January 1829 in Nafplio after the recommendation of captain of the French Artillery Corps, Jean -Henri-Pierre-Augustin Pauzié (1792–1848) and the approval of the first governor of independent Greece, Ioannis Capodistrias (1776–1831) (see [16], p. 125). The translation and the publishing of this school geometry was a remarkable intervention in the Greek mathematics education, as it had serious influence on school geometry subject of Modern Greek education (see [19, 44]). So the following historical question becomes very important: which strong factors did impose that choice, ignoring the legacy of Euclid? It seems that Pauzié suggests such a dimension. And indeed, his political background indicates a deeper root of this approach. On the other hand, Capodistrias with his advisors, and also the Greek intelligentsia, had their own fault. As a graduate of École Polytechnique, it was common for Pauzié to have as his ideal, the famous and high prestige educational institute (see [17]). His mission was incorporated into the French policy for the liberation of the Greeks and the recommendation of the first Greek state, but also to promote French interests. In general, this policy had as a key pillar the organization of the army of collaborating or allied countries. It was this policy that Pauzié met, promoting a moderate model of École Polytechnique for the newly established Central Military School in the small and poor Greece (see [10], p. 150). For the teaching of mathematics, the Pauzié’s model was a quite demanding and innovative one. This can be seen from the relevant manuals (see [17]), that were the course Guides of the corresponding lectures, and these were: (1) Arithmetic and Algebra by Louis-PierreMarie Bourdon (1779–1854), (2) Geometry and Trigonometry by Adrien- Marie Legendre (1752–1833) and (3) Descriptive Geometry by Gaspard Monge (1746– 1818).

3 These

types of books were translations of Razi and Kouma (see [39]).

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In 1828 the translation and adoption of the Bourdon’s Arithmetic was published as the first from these books and in 1829 the Legendre’s Geometry4 followed, while a Greek translation of the Descriptive Geometry by Monge was never published. Ioannis Carandinos [or Carantinos] (1784–1834) undertook these translations according to the personal proposal of the governor, Ioannis Capodistrias. Carandinos, a professor of higher mathematics at the Ionian Academy in Corfu, was familiar with French mathematics as he had studied, in 1823 at the École Polytechnique and used French mathematics textbooks in his teaching and research (see [30], p. 95). The historical interest is not in the process of promotion and adoption of this French geometry in Greek education, but it is in the passive attitude of the Greek politicians, scholars and teachers on this issue, at that time. It is true that the years 1828–1831 were a very critical transitional period in Modern Greek history. And this because geographical boundaries were formed for the first time, and the first institutions of the newly established Greek state were established. It was the time when the national identity of Greeks was rising. Consequently, guidance, options, features, and attitudes in political and cultural events reflect the specificity of the Greek reality at that time. In politics, Capodistrias dominated,5 his primary goal was to secure the support of Russia, France and England towards the recognition of the independence of Greece and the settlement of the borders in favor of Greece. On the other hand, he sought the foundation of a state structure and function, creating a national army, the organization of administrative institutions and the development of educational system (see [4], p. 44). Towards this context, Capodistrias’ educational policy had two goals: the Europeanization of Greek culture and secondly the shaping of the national identity of the Greek people, emphasizing moral-religious education and native tradition (see [29], p. 298). In order to accomplish this difficult merge, he used his centralized method of governance but this did not help him at all. Moreover the educational policy was hard to apply due to the detachment from the intellectuals devoted to Enlightenment, and especially from those contained in the circle of Adamantios Corais (1784–1833).6 As a result, the ideas of the preparatory work, the books and the experiences of progressive teachers of that period such as Benjamin of Lesvos, Theophilos Kairis (1784–1853), Konstantinos Koumas (1777–1836) were not implemented. Instead, he chose to place in key positions in education, biased

4 The Greek translation of this concept was according to the 12th edition of the Legendre’ Géométrie, in 1823. 5 He was an aristocrat (he had the title of Count), educated with medical and philosophical studies in Padua, Italy, and a high degree of political prestige, due to his role in Russian diplomacy as a leading adviser in foreign policy for Tsar. 6 He was great spiritual leader of modern Greece. Through his work in Classical philology, Corais aimed at inculcating in Greeks a sense of their ancient heritage. His emphasis on the need to resurrect Greece’s ancient glory stemmed in large part from his intense hatred of the Orthodox higher clergy whom he blamed for the degraded state of the populace. For Corais, then, the model for a new Greece should be ancient Athens rather than medieval Byzantium (see [11], p. 10).

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and ideologically neutral intellectuals such as his compatriot, Andreas Moustoxidis (1785–1860) and the conservative educator, Ioannis Kokkonis (1796–1864). Considering now the kind of schools that were promoted by the educational policy of Capodistrias it is founded that its main effort was the establishment and operation of mutual learning (Lancasterian) schools in elementary education, which were organized according to French models. Moreover, they had as pedagogical educational background, the corresponding textbook7 of Louis-Charles Sarazin (1797–1865), which was officially translated, introduced and established in 1830, by Kokkonis. If the political will was different, then the responsible ones of these educational method choices could attract and appoint a regulatory role for mathematics education in Theophilos Kairis who was then one of the leading teachers of the nation, tested and recognized in science subjects (see [42]).And it was he who greeted and addressed Capodistrias, when he came to take the first government of Greece in January 1828. This is another fact that at least suggests his affability to the new governor. Moreover, he showed his teaching intentions when a few years later, in 1836, he taught advanced mathematics on his own initiative in his homeland, the island of Andros. However he was ignored, as other personalities of the pre-revolutionary period with similar qualification and experience in teaching mathematics were ignored, and left isolated in foreign countries, such as Konstantinos Koumas, Daniil Filippidis (ca.1750–1832), Stefanos Dougkas (ca.1770–1830), Dimitrios Govdelas (1780– 1831), etc. (see [15]). The worst thing was that Adamantios Corais was ignored, or rather despised. He was a spiritual leader of Modern Greek culture and mentor of the most progressive teachers of prerevolutionary period among who were Kairis, Koumas, Philippidis and more. The statement that “Corais died embittered in 1833, after seeing his works being burned in public in Nafplion from Capodistrias’ followers” (see [23]) reflects the great cultural folly of the political leadership at the time. This is how the chance for a revival of ancient Greek culture in modern education was lost. This was a centerpiece of Corais, along with the emphasis that he gave to the European modernization of the country. The intellectual, ideological and patriotic ideals of the Enlightenment of the pre-revolutionary period were detained. As a consequence, at the origins of mathematics education of the Modern Greek state, the most common mental institution for the revival of Euclid was not benefited. In this educational policy of Capodistrias was raised the relative reactions and criticism. Corais was one of the first ones who expressed his offense. In 1830 he published his oppositions on the educational methods of Capodistrias and of his associates. He denounced the governor that “he hated everything reminding of the ancient glory, ..., that is why he despised ancient monuments and experienced pleasure in their destruction”. Moreover he criticized Moustoxidis because he hated “the Greek culture, the Greek name and antiquities” and that he full of “spirit of

7 Sarazin, L.-Ch.: Manuel des écoles élémentaires, ou Exposé de la method d” enseignement mutual; suivi des ordonnances, arrêtes,règlements concernantl” instruction primaire, Paris, 1829.

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slavery and cowardice, spirit of the illusion ... Jesuitical spirit”(see [29], p.357). Philhellene German scholar Friedrich Wilhelm von Thiersch (1784–1860) had the same opinions. Other intellectuals and politicians criticized various aspects of this educational policy such as Spyridon Trikoupis (1788–1873), George Ludwig von Maurer (1790– 1872) and Henri Dutrône (1796–1867). Also there were opposing teachers, such as Grigorios Konstantas (1758–1844) and Ioannis Venthilos (see [29], p. 355). It is worth saying that there were no qualified mathematicians in the free Greece at the time. The only person who can be described as a mathematician of that time was Ioannis Carandinos who was a professor at the Ionian Academy in Corfu, which was out of the Greek territory at that time. It was him who Capodistrias relied to for the translation of French mathematics books that were required by the Military School. Carandinos apart from translating, he contributed to the education of some teachers who taught mathematics and promoted new textbooks in mathematics, in early modern Greece. Among them were: Dimitrios Despotopoulos, Christos Vafas (1804–1880), Gerasimos Zochios (1811–1881) and Georgios Kondis (1812– 1863) (see [52], p. 202). Students of Carandinos had a validated secular education which consisted of a combination of philosophical and scientific learning, where mathematics was one of the branches of their education. This fact enabled them to teach literature courses, except of scientific lessons. So, they could be employed in education as teachers of higher educational levels, and not only as qualified teachers of mathematics. One such case was Christos Vafas who taught literature courses at the beginning of his teaching career (see [41], p.120). Carandinos’s students were imbued by the teachings of their teacher that French mathematics of the early nineteenth century was dominating. This effect was reflected in the choices most of them made to translate and publish French books on the first decades of the Greek state. Another source of mathematics teachers, who then taught in Greek schools, was the progressive advanced schools who worked ˙ at Smyrna (Izmir, in Turkish) and Kydonies (Ayvalık, in Turkish), before 1821. Such cases were: Nikolaos Chortakis, teacher of mathematics at Central school of Aegina, who had studied at the Philological Gymnasium in Smyrna and Petros Efstratios (1774–1843), teacher at the “Hellenic” school of Tinos, fellow of Kairis in the Academy of Kydonies (see [33], p. 162). Education in these two progressive schools of Asia Minor region reached up to an advanced level of philosophical and scientific knowledge. In the Philological Gymnasium of Smyrna, the teaching of mathematics was raised a good level by Konstantinos Koumas who was influenced by the University of Vienna, where he studied around 1805. At the same time in the Academy of Kydonies, Benjamin of Lesvos and Theophilos Kairis had a dominant role in the teaching of mathematics (see [14]). They were both influenced mainly by French mathematics (see [18]). The institutional and cultural choices of Capodistrias and of his associates along and the available at the time teachers, the teaching of geometry was too weak to nonexistent in primary and secondary education. It was established only in the

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Military School, with the intervention of the French, consolidating this way due to Legendre’s Géométrie. Consequently, the Geometry of Euclid was left away, while the introduction of the relevant foreign Geometry acquired a deep root in the Greek education.

4 The Bavarian Reorganization of Greek Mathematics Culture In 1832 the “Protecting” Powers imposed the 16-year old Prince Otto von Wittelsbach, son of the philhellene King of Bavaria Ludwig I (1786–1868), as the monarch of the newly established Greek state. In January of 1833, the underage Otto, along with a team of consultants and a strong force of Bavarian army came to Nafplion and took the reins of the country (see [4], p. 47). The regency, a three-member board of prominent Bavarians, undertook the difficult task of reconstructing the Greek state until King Otto became an adult. The Regency was composed by Count Joseph Ludwig von Armansperg (1787–1853), Georg Ludwig von Maurer, professor of Law, and the General Karl Wighelm von Heideck (1788–1861). Maurer had undertaken the education. It was him who instituted, in 1834, the first level of Greek education and made a relevant preparation for the other levels. Maurer had in mind the analysis of the Greek educational conditions and the guidelines for its reformed by Thiersch, who had been assigned by Ludwig I. Eventually, Armansperg legislated the secondary and tertiary education in 1836 and 1837 respectively. The spirit and orientation of the eminent Bavarian intellectuals and political managers of the royal governance in Greece, was the inculcation of neoclassicism in the Greek culture at the time. The revival of ancient Greek culture was considered as a crucial dynamic element of Modern Greek education. This is why the scholastic learning of ancient culture and archaizing language was imposed unfavorably of science subjects (see [22], p. 44). Although Bavarian mentors showed emphasis on Greek education in classicism, they did not ignore Euclid. Since 1833, Thiersch noted that “one to approach [mathematics] via studying Greek literature, it is right to consider the Elements of Euclid as basis” (see [43], p.157). Meanwhile, he stressed the pedagogical importance of teaching mathematics that was developed from the ancient Greeks and Archimedes, in particular. Another member of the royal Bavarian administration in Greece, Johann Franz (1804–1851), who participated in the committee for the organization of schools established by the Regency in the spring of 1833, proposed among the basic school books, Euclid and the geometry of Diesterweg (see [1], p. 109). These proposals proved to be spontaneous and superficial, as after their presentation they were not implemented; they were only showing national interest, but these proposal were presented and stayed airy, uninteresting for requirements of their creation. If

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these proposals were realistic and applicable, they would indicate or include ways of implementing and legislating. In contrast, they acted as an enriched rhetorical discourse that helped in the establishment of formal classicism and to the cultivation of a detached historical consideration. The newly created educational structures worked within the new institutional and intellectual conditions. The list of mathematics books published in Greece in the period 1834–1843 provide a revealing example of the mathematics interests and the relevant trends within Greek educational reality. The writing and translation activity was associated directly with the instructional needs of the new educational system. It is also related with some guidance options. On the one hand, there was the promotion of French models and on the other, the German preferences. The students of Carandinos in the Ionian Academy, Despotopoulos, Vafas and Kondis were dedicated to the French mathematics literature, while Soutsos,8 Rizos Ragavis9 and mostly G. Gerakis,10 who studied in Germany, introduced German mathematics culture in Greek education. In this context, Legendre’s Géométrie was republished in 1840 as a second edition of the translation of Carandinos. This means that there was demand for it and this fact reflects its acceptance and validity in the new system of Greek education. In the Military School the teaching of Legendre’s Geometry was preserved, although. Bavarians had made major changes to the educational structure and syllabus (see [17]). This is due to the long lasting French influence, combined with the remaining of D. Despotopoulos, student of Carandinos, in the position of professor of mathematics. In late 1830, a second strong momentum of Legendre’s Geometry in Greek mathematics education was the integration of university courses of the first professor of mathematics, Konstantinos Negris (1804–1880) (see [18], p. 533). But in secondary education he created a certain tendency, with the establishment and creation of high schools,11 where the subject of geometry was included into the

8 Skarlatos Soutsos (1806–1887) was an aristocrat; he studied on scholarship of King Ludwig I at the Military School of Munich. He served as an officer of the Greek army; he joined the Greek military school, worked in politics and served as Minister of War, http://el.wikipedia.org/wiki/∑καρλατoσ_∑o ´ υτσoσ ´ 9 Alexandros Rizos Ragavis (1809–1892) was an aristocrat; he studied at the Military School in Munich where he had the support of King Ludwig I. He first served as an officer in the Greek army, and later as a secretary of the Ministry of Education and Ministry of Interior. He became Professor of Archaeology at the University of Athens. He undertook as a Ministry of Foreign Affairs in the period 1856–1859, and in 1867. Afterwards he served as Ambassador of Greece to the U.S., Istanbul, Paris and Berlin. Important was also his contributions in Greek Literature. 10 George A. Gerakis, was one of the first scholars of the Greek government to study in Germany, circa 1837. Afterwards he taught mathematics in “Hellenic” school and Gymnasium of Patras, from 1841. He published several mathematics textbooks and translated into Greek respective textbooks of F. W.D. Snell and C. Koppe (see [50], p.202]). 11 In the late 1830’s there were three high schools, one in Nafplion, one in Athens and one in Syros island.

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syllabus and two of the first teachers of mathematics were Christos Vafas in Athens, and George Kondis in Nafplio.12 The spread of this French geometry in Greece started in the time when Modern Greek enlightenment was suppressed and detained. The ideology of preRevolutionary struggle for cultural rebirth and the liberation of enslaved Greeks who sought to reconnect the values and ideals with classical roots via the European modernization disoriented towards another kind of acculturation which was the German prototypes, i.e. the orientation towards the German culture of the era, as a contemporary ideal of ancient Greek culture. Georg Ludwig von Maurer strongly supported this ideology noting that: it would be sufficient for the Greeks to imitate the Germans, to become what they have been in the past again (see [28], p. 469). This context favored the selection and spreading of Greek translations via German mathematics textbooks. In this case, Georgios Gerakis systematically promoted this penetration of Greek culture, not only by translating German books in Greek, but also by composing, with a similar spirit, his own textbooks for the teaching of mathematics, that immersed the Greek schools for several decades in the mid-nineteenth century. This intervention was the competitive pole to the French influence, which had long and deep foundations in modern culture. The strange and controversial feature of the cultural engagement of the Bavarians in the Greek education was their weak and in indifferent attitude towards the monumental legacy of Euclid, while King Ludwig I, father of Otto, and all Bavarian intellectuals were infatuated with the monuments of Greek ancient period and with its ideals. Euclid, as it seems, was the great absentee in Greek literature during the first period after the Bavarian establishment of Greek education. Apart from the rare comments by Thiersch and Franz, of 1833, the Greek references to the name and work of Euclid are rare to nonexistent. An unexpected glimpse emerged, in 1857, in a book of university education of Greek students. This was the Guide for Students of the University of Otto by Athanasios Roussopoulos (1823–1898), professor of archaeology, who, among others, presented an interdisciplinary program of university mathematics education, which suggested as an auxiliary class of first year “the study of Greek and Latin classics and in particular of Euclid”, similar was for Archimedes in the second year and Apollonius in the third year. The thought behind this unusual proposal was the disengagement of the mathematics of a self-existence science and the connection between mathematics and other courses such as science, literature and philosophy (see [32], p. 45). This progressive consideration brought forth the work of Euclid, Archimedes and Apollonius as educational content with mathematics, literary and historical value. This idea did not interest the university establishment and the political leaders of the Greek culture at the time. However, as it is showed, the Greek intelligentsia, in the decade of 1850, wasn’t in complete indifference with the legacy of eminent mathematicians of Ancient Greece. It also pointed out that the

12 Before

Kondis, at the Gymnasium of Nauplion were teaching mathematics, Dimitrios Stavridis (1803–1866), who graduated from the Polytechnic of Vienna, and Ch. Vernardos (see [7]).

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cultural and educational context of Greece had very feeble supports for a deeper understanding of national reality and its perspectives. At that time, the teaching of mathematics in Greek university was thematically very limited and scientifically, quite primary. The first 10 years, 1836–1845, the first professor of mathematics, K. Negris, was teaching: Arithmetic, Algebra, Trigonometry, Legendre’s Geometry, Hachette’s Descriptive Geometry and Elements of Calculus. These courses were a very small percentage of the total curriculum of the program for mathematics studies. The non-mathematics courses included teaching other scientific lessons such as Physics, Chemistry, Botany, Zoology and Mineralogy, and several literature courses, including Ancient Greek and Latin authors, Rhetoric, Philosophy, History, Archaeology and Elements of Political Science. The next period, 1847–1853, the variety of courses remained the same, teaching, however, mathematics was limited to the basics, that’s without the Descriptive Geometry and Elements of Calculus. That time, after the removal of Negris due to political reasons, in 1845, George Vouris (1806–1860), professor of astronomy, mathematics and mathematical physics undertook the teaching of mathematics. Vouris had studied in Vienna, where he was the headmaster of the Greek school and in 1841–1843 published five volumes of a mathematics series. It is worth noticing that the fourth volume of the geometry had a synthetic conceptual basis, however, it wasn’t Euclid’s type, or Legendre’s. The organization of the content had its base in: “longimétrie”, planimetry and stereometry, referring to Austrian-German views of school geometry. It seems to be that the geometrical education during the first 15 years of the Greek. University was governed by a blind mimicry of French or German standards. It seems like an attitude that was reflected in the general Greek culture of that period, but also reflected in the cultural and political reality of the newly established Greece. So, French and German influences existed in school mathematics too. In the period of 1854–1873, there was a large increase in students and teachers in secondary schools. Examples noted the development of secondary schools and correspondingly the number of students in four different years, in the following table Fig. 1 (see [25]). The image of the changes also in the lower circle of secondary education is similar, namely Fig. 2. Fig. 1 A table of evolution of students in Public Gymnasium

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Fig. 2 A table of evolution of students in Hellenic schools

These data should include secondary private schools too, which in 1856 were 4, two in Athens and two on the island of Syros. In 1867 were 20 boy’s schools and 7 girl’s schools, these numbers were formed in 1874 in 18 boy’s schools and 15 girl’s schools (see [49], p. 239). This large increase in secondary schools and their student sure cause an equivalent demand for textbooks and therefore school mathematics textbooks. Until 1882, the institutional conditions permitted the existence publication of multiple approved textbooks for the same course, so, the creation of competitive trends authoring or translating manuals for students were normal. In mathematics, these trends were generally two: in one hand, teachers who came from the Ionian Academy of Corfu, with most notable Ch. Vafas, who supported the French manuals and on the other hand, George A. Gerakis who was inspired and promoted the corresponding German spirit. It seems that for both trends were chosen some popular French and German books, which were translated for the needs of secondary schools, while for the. “Hellenic” schools prefer their own compilation or synthesis from textbooks of their respective orientations. So, the geometry of Legendre had already been established in the Greek education, in 1830. On the other hand, in 1840, G. Gerakis juxtapose the German geometry of F.W. D. Snell (1771–1827) and in 1855 he upgraded the translation with the most modern and successful textbooks of Carl Koppe (1803–1874), at first publishing arithmetic and algebra, 2 years later Planimetry and Stereometry in 1858. Towards this dynamic, G. Zochios published his own translation of the Legendre’s Géométrie, the 12th edition of the French, like this of Carandinos. It is worth noting that 17 years had passed from the second edition of Carandinos’ translation, which died in 1834 and in Greece there was no law for publishing or “copyrights”. It is a fact that Zochios translated the 12th edition of the Legendre’s Géométrie again, having the purpose of improving Carandinos’ translation. However, the differences of these translations was in the expressions and had actual no differences. Zochios taught mathematics in previous years in various schools such as the Central School of Aegina and later at a Greek school of Smyrna. In 1856–57 he taught, among other things, the Legendre’s Geometry in the “Elliniko Ekpedeytirio”

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(“Hellenic Educational School”), which was a private school in Athens.13 He started being a writer in 1854, by publishing the Elements of Algebra (1854) and the Essay of Logic Arithmetic (1856). His translation and the edition of Legendre’s Géométrie, in 1857, were reprinted in 1862. Between the 5 years between 1857 and 1862 of Zochios’s two editions of his Legendre translation, there appeared another translation of the same geometry, in 1860. This was the translation of the Legendre’s Géométrie revised by M. A. Blanchet, first published in French in 1845. Translator was Christos Vafas, a respected school teacher, writer and translator of many books for secondary school mathematics education in the period 1856–1861. Moreover he was the owner and director of a private school and in 1861 he became the headmaster of the First Gymnasium of Athens. The plethora of its own mathematics textbooks and translations is comparable only to the works of Gerakis, in the nineteenth century. As it seems, they expressed the two opposing tendencies of the Greek mathematics culture at the time: Vafas as a student of Carandinos, was influenced by France and Gerakis was influenced by Germany. It is possible that Vafas’ translation could be a reaction to of C. Koppe’s Geometry, which was supported by Gerakis in 1857–1858. The truth is that the proposal for this translation to be made was done by Ioannis Papadakis (1825–1876), professor of mathematics and astronomy in the University of Athens, the period 1850–1876 (see [45]). This translation had at least one revision in 1870, which is very revealing because in the new prologue it includes the “proof” of Vafas for the axiom of parallels. This is a rather belated revival of the matter, which Legendre raised himself from the first edition of his Géométrie in 1794 on (see [31]). This Greek approach is one of the few mathematics research attempts,14 in the first decades of modern Greece. This research approach led in 1872 to a mathematics controversy between Vafas and Vassileios Lakon (1830–1900), who was professor of applied mathematics in the University of Athens (see [27, 46]).The controversy and criticism on issues of education, and mathematics education were not rare at that time. Such a comment is noteworthy because it addressed key cognitive and pedagogical aspects of mathematics textbooks while popular writers and translators are indirectly involved. It is about the following statement, made in 1856 by Anthonios Fatseas (1821– 1872): We don’t have a methodical mathematics system in the Central Schools. This is because Mr Vafas” books have no method. On the other hand, the translation of the basic series of Mr. Koppe is a German needless work, inappropriate for us and we have bad connection to it. A student has not only to take mathematics lessons but also others and with no consistency these are not fitting anymore (see [9], p. 34).

These words show the rejective attitude of this teacher towards the most popular textbooks of mathematics. He was a graduate of Theology of the Corfu Ionian

13 See

(in Greek) in Journal Pandora, 8, 1857–58, p. 181. this research was an attempt that can be seen as the second Greek mathematical research effort, with the first one this of Carandinos in the late 1820s.

14 Specifically,

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Academy, who taught mathematics at various schools and published five mathematics textbooks in the period 1851–1870 (see [37]). Although he was not a student of Carandinos but a student of his successor in the Ionian Academy, he was influenced by the French spirit. This is shown by his decision to translate and adopt, in 1858, a textbook of French trigonometry. Likewise, he also made a synopsis of Legendre’s Géométrie, published in 1870.15 The feeling of educational controversy in that period was kept well, as it expressed not only the spirit of corrective comments, advisory exhortations or improvements of some scientific flaws, but sometimes selfish competitions. Such a case was the aggressive criticism, in October 1867, by Lakon in a school Arithmetic textbook by Antonios Damascinos (1836–1905), published that year (see [26]). The response of the affected was immediate and aggressive (see [5]). Unsurprisingly, the controversy continued over the following months, with relevant articles in the newspapers “Alitheia” and “Avgi” to reach an answer-review, page 36, of Damascinos, on January, 1868 (see [6]). It is true that this dialogue revealed some significant aspects of Greek mathematics background and related attitudes of that period. It seems that one of the deepest aspirations of this debate was fierce struggle for supremacy in the market of school mathematic textbooks. Both “defendants” were writers of the entire range of school mathematics books, but also of school physics books for secondary schools, from late 1850 until early 1880. Both were highly qualified and had prominent positions in higher educational institutions. Specifically, the Lakon was the first professor of mathematics in the University of Athens, established in 1848. Later, he attended mathematic courses in Paris for 3 years, in 1854 and he was announced to become an assistant professor of experimental physics at the University of Athens. In 1863 he was promoted to pure and applied mathematics extraordinary professor and in 1868 professor of the same chair, which he held until 1896 (see [38, 51]).On the other hand, Damascinos initially studied mathematics at Athens University and then attended the Sorbonne mathematics courses and took his degree in mathematics from the University of Lyon. He also taught mathematics in the Polytechnical School, in 1877–1884. The translation of Damascinos of Legendre / Blanchet’s Géométrie was not left out of the context of this confrontation. From the words of the translator it is found that he made several interventions to improve the mathematics structure of the French prototype. Characteristic examples are the following assertions of Damascinos for this translation: I added to the prototype findings and quite a few reviews, in the same way I excluded numerous theorems and problems, and I added not only about 25 theorems and problems, but also included the most important Geometry by Mr. Blanchet and Legendre in the IV and VIII of the book changes and I made it perfect (see [6], p. 14).

This translation was first published in 1862 and had at least five renditions: 1865, 1869, 1870, 1872 and 1878. 15 The

original that was used for this summary is unknown.

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Without doubt, this plethoric arrival of Legendre’s Géométrie in Greek mathematics education influenced deeply and perhaps bewitched the first generations of mathematics teachers on newly established Greece. Thus it was established as a powerful referential system of the Greek school geometry for the course of history until today. In this way the partiality of politics and ideology of the Europeanization of the Modern Greek civilization unconsciously and spontaneously buried the glorious heritage of Euclid, Archimedes, Apollonius and other pioneers of the Ancient Greek culture. So the Euclid’s Geometry was left aside. Apart from minimal references they were no effects on the spreading of validated ideals of foreign custodians of the new Greece. In this way any possible Greek interest in the monumental legacy of Euclid was detached. It was a detachment that legalized the relative inaction as a normative background of Greek geometric behavior. Wondering did anyone react? Or the xenomania killed any national consciousness and dignity? Luckily someone stood up. It is about a natural outbreak from a prominent factor in the wider Hellenism. This was Alexandros Caratheodory (1833–1906),16 an important personality of Hellenism in Constantinople and the Ottoman official court. The relative involvement happened in 1863, in an article for Secondary Greek Education, which stated as the following: No I cannot not express my sorrow for the remissness that the teaching method has towards the great Greek scientists. These are who established exact sciences, whose idea is the main issue of scientific studies, and whose names remain buried and forgotten by us. The British, more accurate than us, use now as basic Geometry the Euclid’s Geometry, as other Geometry beyond this does not exist. We should have already been using Geometry of Euclid, even without the complex theories if you want. Instead of this we read terrible translations while we ignore (as students) the existence of Euclid, and so we do for Apollonius, the great author of conic sections, and of Archimedes, whose many theorems are within the frame of the mathematics lessons taken in high school (see [3], p. 187–188).

This discourse is quite structured, nationally characterized and sharp. It criticized Greek negligence on the missing of Euclid and other ancient scientists from Modern Greek education. Also, it mentioned to the prominent position of the geometry of Euclid in English education at the time, as a counterexample of the blind obsession with the modernized knowledge in mathematics education. In other words, it highlighted the major national problem of the Greek mathematics education of that time and recommended the support elements and revealed the importance of incorporating in it the ancient Greek heritage. It is not by luck that this discourse was given in a context independent of external political and cultural influences and impositions of free Greece. The Greek Literary Association of Constantinople, in which the relevant views of A. Caratheodory were published, was a kind of Greek ministry of culture of the broader Hellenism.17 As a cultural institution, this Association has echoed two attitudes: this of the tradition of the

16 He

studied Law in Paris and graduated with a doctorate in 1860. apart from the independent Greece.

17 Greeks,

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Orthodox Patriarchate of Constantinople and that of national self-sensitized. A. Caratheodory18 was inspired by the two attitudes: on the one his family had family ties with the Orthodox Patriarchate of Constantinople and participated in its affairs and on the other he was interested and got involved with the ancient Greek literature, as far as with ethnically Greek issues (see [12], p. 29). Unfortunately, this intervention was isolated and occasional, without continuation or any effects. So, the Greek mathematics education continued its course, unconcerned for the ancient Greek heritage and indifferent to the monumental Geometry of Euclid.

5 Epilogue It was detected that, in the first season of the newly established Greece, the Greek geometrical education remained indifferent to the geometry of Euclid, i.e. between 1828 and 1880. The main factors of this national neglect were two. On the one hand, traditionalism did not favor the development of the ancient Greek heritage and Euclid in particular. On the other hand, spontaneous and “imported”, imitative modernization, cancelled any history or national reflection, which would give meaning to the enclosure of Euclid’s Elements into the latest Greek cultural context. The few fleeting interventions or reactions that favored of Euclid were occasional, isolated and weak, with the effect of “evaporating” away. Euclid was overshadowed by the modernized Legendre’s Géométrie, in the Greek culture. What is the effect of this cultural and educational attitude to the national identity of the Greeks?

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Finite Element Methods with Higher Order Polynomials Konstantina C. Kyriakoudi and Michail A. Xenos

AMS Subject Classification 65N30, 65M60, 76M10

1 Introduction Engineering problems sometimes can be complicated to solve analytical because the partial differential equations that describe them. Thus, numerical methods need to be applied to solve those differential equations. A widely known method for such problems is the finite element method (FEM). The FEM were initially applied mainly in the field of structural mechanics and later the approach was extended to include applications in fluid mechanics, where convective terms play a significant part and contribute to a non-linear formulation of the problem. The non-linearities and instabilities in the solutions to these issues slowed down the progress of the method in fluid mechanics. In the finite element method application, the design of the mesh and the choice of the elements are two of the most important considerations. In the classical approach, piecewise polynomials of fixed degree p are utilized and the mesh size h is decreased for accuracy. This method is called the h-version of FEM. On the other hand, the p-version refers to a fixed mesh and p is allowed to increase and the hp-version combines both the approaches [1, 8–10]. The h-version of the finite element method commonly uses linear or quadratic Lagrangian-interpolant shape functions. The accuracy of this method can be improved while increasing the number of elements in the domain resulting in decreasing the element size [18]. The p-version of the finite element method is a variation of the classical approach where the approximation is represented by higher-order polynomials. The hp-version or Spectral methods benefit from modifications in both the mesh and the polynomial degree.

K. C. Kyriakoudi · M. A. Xenos () Department of Mathematics, University of Ioannina, Ioannina, Greece e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_10

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The review starts with the classical formulation of the finite element method. Then the different variations of FEM are presented, beginning with the p-version and the hp-version. Additionally, convergence is briefly studied for the aforementioned variations. The various types of refinement are then described in the section on adaptive mesh refinement. Some popular fluid mechanics problems are presented in the applications part. Decisively the primary emphasis is on aiming at increasing the polynomial degree of the equations and observing the accuracy of the approach.

2 Finite Element Method Ritz, a German scientist, formulated the fundamental concepts of the FEM in 1909 for approximating solutions for problems with flexible solid mechanics. His method included estimating an energy functional while using known functions multiplied by unidentified coefficients. A system finds those coefficients by the minimization of the functional in relation to each unknown. The functions that can satisfy the provided boundary conditions are constrained. Courant, in 1943, developed Ritz’s method further by introducing linear functions defined over triangular regions. The term Finite Element was introduced in 1960 by Clough [16]. The main concept of FEM is to replace any continuous function with an approximation in a discrete space where a set of polynomials is used to describe them.

2.1 Basic Theory of the h-Version Consider the following boundary value problem ⎧ 2 d u ⎪ ⎪ ⎪ ⎨− dx 2 = f, x ∈ (0, 1) u(0) = 0 ⎪ ⎪ ⎪ ⎩u' (1) = 0

.

(1)

The first step is to multiply both sides with a test function v and integrate. This yields to the weak form of the problem: a(u, v) = (f, v), ∀v ∈ V .

.

(2)

where 

1

a(u, v) =

.

u' (x)v ' (x)dx

(3)

0

The quantities .u, v in the weak form can be scalar or vector functions depending the dimensions.

Finite Element Methods with Higher Order Polynomials

163

Let us define the function space   V = v ∈ L2 (0, 1) : a(v, v) < ∞ and v(0) = 0

.

as the test space, and the space of square integrable functions in .[0, 1] is noted by L2 (0, 1). It can be proved that these functions can create a Banach space. In the case that it is enforced with an inner product, then the space is called Hilbert (and is the same as a .H 1 Sobolev space). The Lax-Milgram theorem ensures existence and uniqueness of the solution for both the variational and the approximation problems [4, 5].

.

Theorem 1 (Lax-Milgram) Given a Hilbert space .(V , (·, ·)), a continuous, coercive bilinear form .a(·, ·) and a continuous linear functional .F ∈ V ' , there exists a unique solution .u ∈ V , such that, a(u, v) = F (v), ∀ v ∈ V .

.

(4)

In higher dimensional problems, the variational form becomes  a(u, v) =

A(x)∇u(x)·∇v(x)+(B(x) · ∇u(x)) v(x)+C(x)u(x)v(x) dx

.

(5)

Ω

where .A, B, C are bounded and measurable functions on .Ω ⊂ Rn and .B is a vector. The uniqueness of the solution, in this case, is guaranteed [4, 16].

2.2 Shape Functions Due to the utilization of Sobolev spaces, some functions are discontinuous. Therefore, there is a need to focus on Piecewise Polynomial Spaces. Let a partition of .[a, b] with n elements .0 = x0 < x1 < . . . < xn = 1 and let .Vh be a linear space of functions v such that: • .v ∈ C 0 ([0, 1]) • .v |[xi−1 ,xi ] is a linear polynomial and • .v(0) = 0 All .φi functions can be defined, for all .i = 1, . . . , n and .φi (xj ) = δij , Kronecker’s delta. The purpose of this space is to construct an orthonormal basis .{φi : 1 ≤ i ≤ n} for the .Vh space. This is called nodal basis and the points .xi are called nodes. The discrete space creates the necessity to describe the functions using a proper basis. The fact that the coordinates of each element can periodically make the task more complex demands finding a more effective technique to perform our

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calculations. Thus, the following index is provided to overcome those challenges, by transferring the global system to the local, in the interval .[0, 1]: i(e, j ) = e + j − 1,

.

The discrete form of the functions involved in the problem (2) is presented in the following definitions. Definition 1 Given .v ∈ C 0 ([0, 1]), then .vI ∈ Vh is the interpolant of .v and is determined by vI :=

n

.

v(xi )φi .

i=1

In the same fashion, the interpolant of f , .fI , is defined : fI :=

1

.



f xi(e,j ) φje

(6)

e j =0

where .{φje : j = 0, 1} is the basis of the interval .Ie = [xe−1 , xe ] : φje (x) = φj ((x − xe−1 )/(xe − xe−1 ))

.

and φ0 (x) =

.

1 − x,



x ∈ [0, 1]

0,

φ1 (x) =

else

x,

x ∈ [0, 1]

0,

else

At last the bilinear form .a(u, v) is converted to: a(u, v) =



.

ae (u, v)

e

where .ae (u, v) is the local bilinear form in each element defined by the following 

u' v ' dx

ae (u, v) := Ie

.

= (xe − xe−1 )−1



1

0

= (xe − xe−1 )−1



⎛ ⎞' ⎛ ⎞' ⎝ ui(e,j )φj ⎠ ⎝ vi(e,j )φj ⎠ dx j

ui(e,0) ui(e,1)

j

t

 K

vi(e,0) vi(e,1)

 ,

Finite Element Methods with Higher Order Polynomials

165

where K is the local stiffness matrix 

1

Ki,j :=

.

0

' φi−1 φj' −1 dx, i, j = 1, 2.

The solution to the problem results from solving the above system.

2.3 p-Version and Hierarchical Basis The classical form of the finite element method utilizes low-order polynomials such as the Lagrange, to create the basis which is called the standard, whereas in the pversion the degree of polynomials can increase using orthogonal polynomials such as Legendre or Chebyshev polynomials. The resulting base is called hierarchical. The main difference with respect to the Lagrange polynomials, commonly used in low-order finite elements, consists of accounting for the lower-order basis functions when higher-order shape functions are computed [17, 18, 22] (Fig. 1). The selection of shape functions is crucial in defining the finite element space. However, there are several factors to consider when choosing shape functions according to Szabó [21]. To begin with, the shape functions should have as small of an error as possible when mapped to increasing polynomial degrees. It is important for shape functions to allow efficient calculations of the stiffness matrices and permit minimal continuity error. Also, shape functions selection can greatly impact the implementation of iteration procedures, especially for larger problems. Therefore, polynomials with specific orthogonality properties should be preferred for the construction of the shape functions. These properties are to be hierarchic, this means that the set of polynomial degree .p + 1 should contain the set of shape functions of polynomial degree p [2]. With this in mind, the number of shape functions should be as small as possible, especially at vertices, edges, and faces. Fig. 1 Hierarchical basis from Legendre polynomials

Hierarchical Basis Polynomials

2 1.5 1

Pn(x)

0.5 0 –0.5 N1 N2

–1

N3 N4

–1.5 –2 –2

N5 N 6

–1.5

–1

–0.5

0

0.5

1

1.5

2

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The hierarchic shape functions are the integrated Legendre shape functions defined as: 1 (1 − ξ ) 2 1 N2 (ξ ) = (1 + ξ ) 2

N1 (ξ ) =

.

Ni (ξ ) = φi−1 (ξ ), i = 3, 4, . . . p + 1 where .φi−1 is computed using the expression:  φj (ξ ) =

.

2j − 1 2



ξ

−1

Lj −1 (x) dx

 1 Lj (ξ ) − Lj −2 (ξ ) j = 2, 3, . . . =√ 4j − 2 where .Lj (ξ ) are the Legendre polynomials. Since they are computed by means of the Legendre polynomials, they follow its orthogonality property: 

1

.

−1

dNi dNj dξ = δij i ≥ 3 and j ≥ 1 or vice-versa dξ dξ

(7)

The first five degrees of hierarchical shape functions for one-dimensional problems are the following (Fig. 1): 1 (1 − ξ ) 2 1 N2 (ξ ) = (1 + ξ ) 2 1√ 2 N3 (ξ ) = 6(ξ − 1) 4

N1 (ξ ) =

.

1√ 10(ξ 2 − 1)ξ 4 1√ N5 (ξ ) = 14(5ξ 2 − 6ξ + 1) 16 1√ N6 (ξ ) = 2(7ξ 2 − 10ξ + 3)ξ 16 N4 (ξ ) =

The first two shape functions .N1 , N2 are called nodal shape functions or nodal modes. The next shape functions .Ni (ξ ), i = 3, 4, . . . are called internal shape functions, internal modes, or bubble modes, because .Ni (−1) = Ni (1) = 0. The main difference here is that all the shape functions of lower order are embedded in the hierarchical basis. Thus, the functions can be written similarly to before but with the hierarchical basis functions. The bilinear form is computed from 2 .a1 (u, v) = xe+1 − xe



+1 −1

⎛

p e +1

κ(Qe (ξ )) ⎝

j =1

⎞⎛ ⎞ p e +1 dNj dN i ⎠⎝ ⎠ dξ aj bi dξ dξ i=1

Finite Element Methods with Higher Order Polynomials

=

p e +1 p e +1

167

kije aj bi

i=1 j =1

= bT K e a where e .kij



2 = xe+1 − xe

+1

κ(Qe (ξ ))

−1

dNi dNj dξ dξ dξ

The stiffness matrix K is symmetric following the symmetry of the bilinear form and the fact that the same basis is used for .u, v. Following the same procedure the second term of the bilinear form is given from: xe+1 − xe .a2 (u, v) = 2 =

p e +1 p e +1



+1 −1

⎛

p e +1

c(Qe (ξ )) ⎝

⎞⎛ aj Nj ⎠ ⎝

p e +1

j =1

⎞ bi Ni ⎠ dξ

i=1

meij aj bi

i=1 j =1

= bT M e a where e .mij

xe+1 − xe = 2



+1

−1

c(Qe (ξ ))Ni Nj dξ

When using hierarchic shape functions, the stiffness matrix tends to become almost perfectly diagonal [7, 20, 25]. The right-hand side of the equation involves the numerical evaluation of .f (v) xe+1 − xe .f (v) = 2 =

p e +1



+1

−1

⎛

p e +1

f (Qe (ξ )) ⎝

⎞ bie Ni ⎠ dξ

i=1

bie rie

i=1

where e .ri

xe+1 − xe = 2



+1 −1

f (Qe (ξ ))Ni (ξ )dξ

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In the two-dimensional case, three kinds of shape functions are involved. These are nodal, edge, and internal shape functions. Accordingly, elements in three dimensions are characterized by one more group of shape functions, the face nodes [3, 20, 21].

2.4 Spectral and hp Methods There is also a formulation where the mesh and the degree of the polynomials can change [10]. Spectral methods, which were introduced by Patera [15], combine both of the previous versions of the Finite Element Method. Spectral Element Methods benefit from the geometric flexibility of FEM and the convergence rates from spectral techniques [11, 12]. In the higher-order methods, spectral element methods and hp-version of FEM have been included as well as the p-version of FEM mentioned before. Higher-degree polynomials are preferred for their accuracy in smooth problems, thus this approach succeeds the geometrical flexibility in comparison with other spectral methods. The spectral FEM provides better accuracy for a fixed number of degrees of freedom. Finally, the stiffness matrix tends to be diagonal whereas in the h-version is full [18].

2.5 Convergence and Error Estimates In terms of convergence, it has been proved that in p- and hp-versions the rate is significantly better than the classical h-version, especially in smooth problems [2, 8– 10]. More specifically in the h-version, convergence is limited by the degree of the polynomial where it is fixed. When the mesh is designed to be optimal the rate can be higher. The h-version of the finite element method can never achieve a convergence rate better than algebraic, independent of the mesh utilized, whereas the p-, h-p version has the potential to yield exponential convergence rates [6, 7, 18]. For the three cases mentioned before the error estimates have the following forms. When a sequence of meshes is produced using uniform refinement, the estimate in the h-version is 1

||e||E ≤ C1 N − 2 min(p,λ) ≈ C1 hmin(p,λ) ,

.

(8)

where .C1 is a positive constant, N is the number of degrees of freedom p is the polynomial degree, h is the element size and .λ represents the ‘strength’ of the singularities. In the p-version where the mesh is fixed and the degree of elements changes the estimate is ||e||E ≤ C2 N −λ ,

.

(9)

Finite Element Methods with Higher Order Polynomials

169

In a similar manner, for the hp-version the estimate is ||e||E ≤ C3 e−γ N , θ

.

(10)

where .γ , θ are constants. In this case, the rate of convergence is exponential [19, 24].

2.6 Adaptive Sometimes, accuracy problems arise in particular areas of the grid or mesh. Rather than refining the entire region uniformly, it is more effective to refine only the regions that require additional precision. This is called Adaptive Mesh Refinement (AMR), which has a variety of applications in engineering. AMR can be divided into three main categories: h-refinement, p-refinement, and r-refinement. In h-refinement, the type of elements is the same in the domain but the number and size change based on the geometry. In p-refinement, higher-order polynomials are used as the basis for the shape functions, while the mesh element size is constant. In r-refinement, the main feature is that the nodes are relocated in areas that require optimization. The number of nodes and elements here has not been altered. The aforementioned methods can be used in conjunction with one another, such as hp-refinement, hrmethod etc. [14, 23, 24].

3 Applications in Fluid Mechanics This section is dedicated to applications. In the following problems, the main focus is on the effects of the increase in the degree of the polynomials. More specifically, to demonstrate how the mesh changes as the degree of the polynomial increases, adaptive mesh refinement is used. The following results were obtained utilizing FEniCS project [13].

3.1 The Poisson Equation Assume the following problem .

where

−Δu = f,

in Ω

u = uD , on ∂Ω

(11)

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Fig. 2 Initial mesh with 128 elements

Fig. 3 Adaptive meshes with (a) 57,110, (b) 234 and (c) 128 elements

uD =

1,

x=0

0,

x=1

.

The weak form of the finite element method is:   . ∇u∇v = f v. Ω

a(u, v) = (f, v) (x−0.5)2 +(y−0.5)2

(12)

Ω

(13)

0.02 In this problem .f (x, y) = 10e− . The domain is a unit square with 128 elements in the initial mesh, as it is shown in Fig. 2. In each case, the initial mesh has 128 elements. It can be easily seen that as the order of the polynomials is increased, the adapted mesh needs fewer elements for the solution without sacrificing the velocity results (Fig. 3). In the third picture (c) the adapted mesh is the same as the initial, the only difference is the increase of the degree from 1 to 3. The exact number of elements in meshes is 57,110 when there are first-order polynomials, 234 for second-order, and 128 for third-order polynomials (Table 1). The error in both the original case as well as in the adapted case decreases as the order increases as it is shown in Table 1. The numerical solution of the problem is shown in Fig. 4.

Finite Element Methods with Higher Order Polynomials Table 1 Error in .L2 norm

171 Error k=1 k=2 k=3

Initial mesh 1.72055 1.71829 1.71824

Adapted mesh 1.716084 1.716344 1.718240

Fig. 4 Numerical solution of the Poisson equation

2.0e-01 0.15 0.1 0.05 0.0e+00

3.2 The Stokes Equation The following problem is the Stokes equations ⎧ ⎪ ⎪ ⎨−νΔu + ∇p = f in Ω . ∇ · u = 0 in Ω ⎪ ⎪ ⎩ u = 0 on ∂Ω.

(14)

where .ν denotes the kinematic viscosity and .Ω ⊂ R2 is a bounded domain. The function .u denotes the velocity and p the pressure. The corresponding weak form of the Stokes equation is ν(∇u, ∇v) − (∇ · v, p) = (f, v).

.

(∇ · u, q) = 0

(15) (16)

In the cases studied below a mixed function space .W = V × Q is utilized with Taylor-Hood Elements. Taylor-Hood is a mixed element containing the .(Pk , Pk−1 ) pair of polynomials with .k ≥ 2.

3.2.1

Backward Facing Step

A well-known test problem of the Stokes problem, for internal flows, is the Backward Facing Step. Due to the geometry, it creates a recirculation zone near the wall of the step.

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The problem has the following formulation ⎧ ⎪ ⎪ ⎨ −νΔu + ∇p = f in Ω .

⎪ ⎪ ⎩

∇u · n + pn = g in 𝚪N

(17)

u = uD on 𝚪D .

where .𝚪D , 𝚪N are the Dirichlet and Neumann boundary conditions, respectively. The weak form is :    . f · vdΩ + g · vds [ν∇u · ∇v − (∇ · v)p + (∇ · u)q] dΩ = Ω

Ω

𝚪N

a ((u, p), (v, q)) = L(v, q) Initially, the domain has 890 elements (Fig. 5), and the adapted mesh with the (P2 , P1 ) space has 1131 elements. As the polynomial order increases the number of elements decreases (Fig. 6). Mesh tends to initial one when .k = 10 and .((P10 , P9 )) polynomials are used (Table 2). The error in this problem is decreasing as the number of elements tends to the initial number of elements. Fluctuations in errors occur because in some cases there is difficulty in reducing the number of elements, resulting in either the number remaining constant or increasing. In all cases the solution is the same as in Fig. 7.

.

Fig. 5 Initial mesh with 890 elements

Fig. 6 Adaptive Meshes with (a) 1131, (b) 1093 and (c) 909 elements

Finite Element Methods with Higher Order Polynomials Table 2 Error in .L2 norm

Error (P.2 , P.1 ) (P.3 , P.2 ) (P.4 , P.3 ) (P.5 , P.4 ) (P.6 , P.5 ) (P.7 , P.6 ) (P.8 , P.7 ) (P.9 , P.8 ) (P.10 , P.9 )

173 Initial mesh 0.912279 0.911585 0.911806 0.911766 0.911816 0.911788 0.911805 0.911787 0.911796

Adapted mesh 0.911745 0.911762 0.911780 0.911765 0.911774 0.911769 0.911775 0.911778 0.911796

Fig. 7 Numerical solution of the backward step

3.2.2

Lid Driven Cavity

The Lid Driven Cavity problem is another benchmark problem that is used in fluid mechanics. This problem is mainly studied due to the fact that exhibits a variety of phenomena that occur in incompressible flows such as secondary flows, complex flow patterns, turbulence, etc. There is a square domain with three rigid walls with no-slip conditions and a moving lid. In this problem, velocity is set to be equal to 1. .

−νΔu + ∇p = f in Ω ∇u = 0 in 𝚪N

(18)

In the beginning, the mesh has 128 (Fig. 8) and the adapted one 4669 elements, but as the order of the polynomials is increased the number of elements in the adapted mesh tends to become the same as the starting one (Fig. 9). As in the previous cases, a reduction of the error is observed as the degree of the polynomials increases. In some cases increasing the degree led to an increase in the mesh elements and this can be observed also in the errors table (Table 3). The yielding solution can be seen in Fig. 10.

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Fig. 8 Initial mesh with 128 elements

Fig. 9 Adaptive meshes with (a) 4669, (b) 391 and (c) 173 elements Table 3 Error in .L2 norm

Error (P.2 , P.1 ) (P.3 , P.2 ) (P.4 , P.3 ) (P.5 , P.4 ) (P.6 , P.5 ) (P.7 , P.6 )

Initial mesh 0.75425947 0.75317886 0.75317088 0.75307177 0.75314475 0.75310587

Adapted mesh 0.75309220 0.75308080 0.75309430 0.75309132 0.75312226 0.75307011

Fig. 10 Numerical solution of the lid driven cavity problem

1.0e+00 0.8 0.6 0.4 0.2 0.0e+00

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175

4 Conclusions Utilizing high-order elements in the finite element method offers advantages, including providing more precise solutions than low-order elements, while simultaneously using fewer elements. This leads to a more efficient solution process. In this review, the basic theory of the finite element has been presented by analysing the shape functions. The three alternate versions of FEM controlling the mesh and the polynomial degree were discussed, mainly focusing on higherorder polynomials and the p-version of FEM. In the applications section fluid mechanics problems were studied. It can be easily pointed out that the increase in the polynomial degree leads to the solution but with the minimum number of elements possible. This happens since higher-order polynomials can better approximate the functions.

References 1. I. Babuška, M. Suri, The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36(4), 578–632 (1994) 2. I. Babuška, B.A. Szabo, I.N. Katz, The p-version of the Finite Element Method. SIAM J. Numer. Anal. 18(3), 515–545 (1981) 3. I. Babuška, M. Griebel, J. Pitkäranta, The problem of selecting the shape functions for a p-type finite element. Int. J. Numer. Methods Eng. 28(8), 1891–1908 (1989) 4. S.C. Brenner, L.R. Scott, L.R. Scott, The Mathematical Theory of Finite Element Methods, vol. 3 (Springer, Berlin, 2008) 5. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15 (Springer Science & Business Media, Berlin, 2012) 6. P. Carnevali, R. Morris, Y. Tsuji, G. Taylor, New basis functions and computational procedures for p-version finite element analysis. Int. J. Numer. Methods Eng. 36(22), 3759–3779 (1993) 7. M.O. Deville, P.F. Fischer, E. Mund, et al., High-Order Methods for Incompressible Fluid Flow, vol. 9 (Cambridge University Press, Cambridge, 2002) 8. W.-Z. Gui, I. Babuška, The h, p and hp versions of the Finite Element Method in 1 Dimension: Part I. The error analysis of the p-version. Numer. Math. 49(6), 577–612 (1986) 9. W.-z. Gui, I. Babuška, The h, p and hp versions of the Finite Element Method in 1 Dimension: Part II. The error analysis of the h-and hp versions. Numer. Math. 49, 613–657 (1986) 10. W.-z. Gui, I. Babuška, The h, p and hp versions of the Finite Element Method in 1 Dimension: Part III. The adaptive hp version. Numer. Math. 49, 659–683 (1986) 11. G. Karniadakis, S. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics (Oxford University Press on Demand, Oxford, 2005) 12. H. Lee-Wing, A.T. Patera, A legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows. Comput. Methods Appl. Mech. Eng. 80(1–3), 355– 366 (1990) 13. A. Logg, K.-A. Mardal, G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, vol. 84 (Springer Science & Business Media, Berlin, 2012) 14. R.H. Nochetto, K.G. Siebert, A. Veeser, Theory of adaptive finite element methods: an introduction, in Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday (Springer, Berlin, 2009), pp. 409–542

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15. A.T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984) 16. A. Raptis, K. Kyriakoudi, M.A. Xenos, Finite element analysis in fluid mechanics, in Mathematical Analysis and Applications (Springer, Berlin, 2019), pp. 481–510 17. P. Solin, K. Segeth, I. Dolezel, Higher-Order Finite Element Methods (Chapman and Hall/CRC, Boca Raton, 2003) 18. M. Sprague, T. Geers, Legendre spectral finite elements for structural dynamics analysis. Commun. Numer. Methods Eng. 24(12), 1953–1965 (2008) 19. B.A. Szabo, Estimation and control of error based on p-convergence. Tech. rep. Washington University ST Louis MO Center for Computational Mechanics (1984) 20. B. Szabó, I. Babuška, Finite element analysis: method, verification and validation, (John Wiley & Sons, 1991) 21. B. Szabó, A. Düster, E. Rank, The p-version of the Finite Element Method, in Encyclopedia of Computational Mechanics, (Wiley Online Library, 2004) 22. H. Xu, C.D. Cantwell, C. Monteserin, C. Eskilsson, A.P. Engsig-Karup, S.J. Sherwin, Spectral/hp element methods: recent developments, applications, and perspectives. J. Hydrodyn. 30, 1–22 (2018) 23. Y. Zhao, X. Zhang, S.L. Ho, W. Fu, An adaptive mesh method in transient finite element analysis of magnetic field using a novel error estimator. IEEE Trans. Magn. 48(11), 4160– 4163 (2012) 24. J. Zhu, O. Zienkiewicz, Adaptive techniques in the finite element method. Commun. Appl. Numer. Methods 4(2), 197–204 (1988) 25. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals (Elsevier, Berlin, 2005)

On Local Asymptotics for Orthonormal Polynomials Eli Levin and D. S. Lubinsky

1 Introduction: Compact Support in R The idea of “local limits” is most easily understood in the context of measures on a compact interval. (Those familiar with the topic, may skip to Sect. 4 to see the new results.) Let .μ be a finite positive Borel measure on the real line with compact support, and with infinitely many points in its support. Then we may define orthonormal polynomials pn (μ, x) = γn x n + ..., γn > 0,

.

n = 0, 1, 2, ... satisfying the orthonormality conditions

.

 pn (μ, x) pm (μ, x) dμ (x) = δmn .

.

We denote their zeros by xnn < xn−1,n < ... < x1n < ∞.

.

E. Levin Mathematics Department, The Open University of Israel, Raanana, Israel D. S. Lubinsky () School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_11

177

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E. Levin and D. S. Lubinsky

The reproducing kernel Kn (μ, x, y) =

n−1 

.

pk (μ, x) pk (μ, y)

k=0

and its normalized cousin n (μ, x, y) = μ' (x)1/2 μ' (y)1/2 Kn (μ, x, y) K

.

play an important role in analyzing orthogonal polynomials. The behavior of .pn as .n → ∞, is a central topic in orthogonal polynomials. Essentially for z away from the support, .pn (z) exhibits geometric growth. Inside the interval of orthogonality, there is oscillatory behavior. As an example, let the support be .[−1, 1] and let .μ' satisfy the Szeg˝o condition 

1

.

−1

log μ' (x) dx > −∞. √ 1 − x2

Then for .z ∈ C\ [−1, 1], .

n   lim pn (μ, z) / z + z2 − 1 = G (z) ,

n→∞

' where G is a function analytic √ in .C\ [−1, 1], involving the “Szeg˝o function” for .μ [7, 26, 27, 29]. Here .z + z2 − 1 is the conformal map of the exterior of .[−1, 1] onto the exterior of the unit ball. Under additional conditions on .μ' , such as a local Dini condition for the modulus of continuity of .μ' , there are asymptotics that reflect the oscillatory behavior for .x = cos θ ∈ (−1, 1): '

pn (μ, x) μ (x)

.

1/2

 1/4  2 2 cos (nθ + g (θ )) + o (1) , 1−x = π

with an appropriately defined function g [2, 7, 29]. The behavior near the endpoints of the interval is more delicate [1]. There is a very extensive literature on asymptotics of varying strengths and generality. See for example [22, 23, 26–28, 32]. There is a gap between the exterior asymptotics and those inside the support: one needs to stay a positive distance from the support to have the former. In a recent paper [13], the second author used universality limits to prove a “local limit”, a type of ratio limit that holds in the complex plane close to the support. Here is a typical example: Theorem 1.1 Assume that .μ is a regular measure with compact support .supp [μ]. Let I be a closed subinterval of the support in which .μ is absolutely continuous, and .μ' is positive and continuous. Let J be a compact subset of .I o and .yj n ∈ J be

On Local Asymptotics for Orthonormal Polynomials

179

a zero of .pn' . Then   pn μ, yj n + nω zy ( )

j n = cos π z . lim n→∞ pn μ, yj n

(1.1)

uniformly for .yj n ∈ J and z in compact subsets of .C. Here .ω is the equilibrium density, in the sense of the potential theory, for the support of .μ. Let us expand on these hypotheses. We say that .μ is regular (in the sense of Stahl, Totik, and Ullmann) if for every sequence of polynomials .{Pn } with degree .Pn at most .n,  .

lim sup n→∞

|Pn (x)|

1/2 |Pn |2 dμ

1/n ≤1

for quasi-every .x ∈ supp [μ] (that is except in a set of logarithmic capacity 0). An equivalent formulation involves the leading coefficients .{γn } of the orthonormal polynomials for .μ : .

1/n

lim γn

n→∞

=

1 , cap (supp [μ])

where cap denotes logarithmic capacity. If the support consists of finitely many intervals, a sufficient condition for regularity is that .μ' > 0 a.e. in each subinterval. However much less is needed, and there are pure jump measures and singularly continuous measures that are regular [28]. The equilibrium measure for the compact set supp.[μ] is the probability measure that minimizes the energy integral   .

log

1 dν (x) dν (y) |x − y|

amongst all probability measures .ν supported on supp.[μ]. If I is an interval contained in supp.[μ], then the equilibrium measure is absolutely continuous in I , and moreover its density, which we denoted above by .ω, is continuous in the interior o .I of I [25, p.216, Thm. IV.2.5]. An alternative formulation of (1.1) is   z pn μ, yj n + ˜ Kn (μ,yj n ,yj n )

. lim = cos π z n→∞ pn μ, yj n

(1.2)

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The essential feature of (1.1) is that it holds for z in a small complex neighborhood of .yj n , hence the name local limit. The proof of the limit involves normal families of analytic functions, and universality limits. There is a close relationship between this asymptotic and “clock spacing” of zeros of orthogonal polynomials. From (1.1) and Hurwitz’ Theorem, it is clear that 1+o(1) near .yj n , the spacing between successive zeros is . nω . In the case of .(−1, 1), (yj n )  π 1−yj2n

this spacing becomes . (1 + o (1)). After the transformation .xj n = cos θj n , n the spacing between successive .θj n becomes . πn (1 + o (1)). When projected onto iθ .e j n , this yields equispaced points around the unit circle. Hence the name “clock spacing”. This has been studied intensively by many researchers in orthogonal polynomials [5, 27, 30, 31] again under varying hypotheses, and with varying levels of generality. The limit (1.1) may be thought of as emphasizing “clock spacing”. A perhaps more impressive application of local limits is to asymptotics at the endpoints of the interval of orthogonality. It is more difficult to establish asymptotics of orthogonal polynomials at endpoints [1], and they are generally available under quite restrictive hypotheses. They involve .Jα , the usual Bessel function of the first kind and order .α, Jα (z) =

∞  (−1)n (z/2)2n+α

.

n=0

n!𝚪 (α + n + 1)

.

(1.3)

and its normalized form, Jα∗ (z) = Jα (z) /zα = 2−α

.

∞  (−1)n (z/2)2n n!𝚪 (α + n + 1)

(1.4)

n=0

The second author proved [14] a local limit of the form: Theorem 1.2 Assume that .μ is a regular measure with support .[−1, 1]. Assume that for some .ρ > 0, .μ is absolutely continuous in .J = [1 − ρ, 1], and in J , its absolutely continuous component has the form .w(x) = h (x) (1 − x)α , where .α > −1, and h has a positive limit at 1. Then uniformly for z in compact subsets of .C, we have   z2 pn μ, 1 − 2n 2 J ∗ (z) = α∗ . . lim (1.5) n→∞ pn (μ, 1) Jα (0) While this is a ratio asymptotic, and the behavior of .pn (1) is not specified, it is still impressive because full asymptotics of orthogonal polynomials at 1 require either specific asymptotics about recurrence coefficients, or substantial local and global smoothness of .μ' .

On Local Asymptotics for Orthonormal Polynomials

181

2 The Unit Circle Let .μ be a finite positive Borel measure on .[−π, π ) (or equivalently on the unit circle) with infinitely many points in its support. Then we may define orthonormal polynomials ϕn (z) = κn zn + ..., κn > 0,

.

n = 0, 1, 2, ... satisfying the orthonormality conditions

.

.



1 2π

π

−π

ϕn (z) ϕm (z)dμ (θ ) = δmn ,

where .z = eiθ . For such measures, the notion of regularity becomes .

1/n

lim κn

n→∞

= 1,

since the unit circle has logarithmic capacity 1. This is true if for example .μ' > 0 a.e. in .[−π, π ), but there are pure jump and pure continuous measures  singularly n that are regular. We denote the zeros of .ϕn by . zj n j =1 . They lie inside the unit circle, and may not be distinct. The nth reproducing kernel for .μ is Kn (z, u) =

n−1 

.

ϕj (z) ϕj (u).

j =0

Local limits for the unit circle turn out to be more difficult, because there is no obvious analogue of the point 1 at the endpoint of .[−1, 1], or the local maximum point .yj n of .|pn | inside the support. The derivative .ϕn' of the orthonormal

 polynomial .ϕn has all its zeros inside the unit circle. Moreover, .ϕn eiθ  might have only a few local maxima for .θ ∈ [−π, π]. In [15], we proved: Theorem 2.1 Assume that .μ is a regular measure with the unit circle as support. Assume that J is a closed subarc of the unit circle such that .μ is absolutely continuous and .μ' is positive and continuous in J . Let .J1 be a subarc of the (relative) interior of J . Let .{zn }n≥1 be a sequence in .J1 . For .n ≥ 1, we can choose at least one of .ζn = zn or .ζn = zn eiπ/n such that from any infinite sequence of positive integers, we can extract a further subsequence .S such that uniformly for u in compact subsets of .C,

ϕn ζn 1 + un = eu + C(eu − 1) . lim ϕn (ζn ) n→∞,n∈S

(2.1)

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where  C=

.

lim

n→∞,n∈S

 ζn ϕn' (ζn ) −1 . n ϕn (ζn )

(2.2)

Moreover, .|C| ≤ 1. We note that there was a (fixable) mistake in the proof of Theorem 2.1 in [15]. The mistake was corrected in [16]. It is obviously of interest to consider the case .C = 0. This turns out to require much more on the orthonormal polynomials than in the case of measures on .[−1, 1]. In [17, Theorem 2.1], we established: Theorem 2.2 Assume that .μ is a regular measure with the unit circle as support. Assume that J is a closed subarc of the unit circle such that .μ is absolutely continuous and .μ' is positive and continuous in J . The following are equivalent: (I) Uniformly for z in proper subarcs of J , and for u in compact subsets of .C,

ϕn z 1 + un = eu . . lim n→∞ ϕn (z)

(2.3)

(II) Uniformly for z in proper subarcs of J ,

ϕn ze±iπ/n = −1. . lim n→∞ ϕn (z)

(2.4)

(III) Uniformly for z in proper subarcs of J , .

lim |ϕn (z)|2 μ' (z) = 1.

n→∞

(2.5)

Thus a full local limit such as (2.3) requires the pointwise asymptotics (2.5) of the absolute values of the orthogonal polynomials.

3 Local Limits for Varying Exponential Weights   The Hermite weight .W (x) = exp − 12 x 2 , .x ∈ R, is probably the first exponential weight whose orthogonal polynomials were thoroughly investigated. Studies of the moment problem and weighted approximation led to consideration of more general exponential weights. It was Geza Freud and later Paul Nevai that began a systematic study of the orthogonal polynomials for exponential weights such as α .W (x) = exp (− |x| ), .α > 1, in the 1970s. The introduction of potential theory, and later Riemann-Hilbert techniques permitted a dramatic expansion of the precision

On Local Asymptotics for Orthonormal Polynomials

183

of analysis, and the classes of weights. See [3, 4, 9, 20, 23] for some historical perspectives and references. Some of the interest in exponential weights arises from random matrix theory. There, rather than considering a fixed exponential weight, one considers a sequence of weights, such as .e−nQ(x) or even .e−nQn (x) [3]. These are called varying exponential weights. As with measures with compact support, potential theory plays a key descriptive role, though here it involves what are called external fields [25]. Let .∑ be a closed set on the real line, and .e−Q be a continuous function on .∑. If .∑ is unbounded, we assume that .

lim

|x|→∞,x∈∑

(Q (x) − log |x|) = ∞.

Associated with .∑ and Q, we may consider the extremal problem   .

inf

log

ν

1 dν (x) dν (t) + 2 |x − t|



 Q dν ,

where the inf is taken over all positive Borel measures .ν with support in .∑ and ν (∑) = 1. The inf is attained by a unique equilibrium measure .ωQ , characterized by the following conditions: let

.

 V ωQ (z) =

.

log

1 dωQ (t) |z − t|

denote the potential for .ωQ . Then V ωQ + Q ≥ FQ on ∑;

.

  V ωQ + Q = FQ in supp ωQ . dω

Here the number .FQ is a constant. We let .σQ (x) = dxQ . See [25] for a comprehensive treatment of potential theory for external fields. In [12, Theorem 2.1, p. 4] we proved: Theorem 3.1 Let .e−Q be a continuous non-negative function on the set .∑, which is assumed to consist of at most finitely many intervals. If .∑ is unbounded, we assume also .

lim

|x|→∞,x∈∑

(Q (x) − log |x|) = ∞.

Let h be a bounded positive continuous function on .∑, and for .n ≥ 1, let dμn (x) = h (x) e−2nQ(x) dx.

.

(3.1)

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  Let J be a closed interval lying in the interior of .supp ωQ , where .ωQ denotes the equilibrium measure for Q. Assume that .ωQ is absolutely continuous in a neighborhood of J , and that .σQ and .Q' are continuous in that neighborhood, while .σQ > 0 there. Then uniformly for .yj n ∈ J that is a local maximum of −nQ(·) and for z in compact subsets of the plane, we have .|pn (μn , ·)| e   z nQ' (yj n ) pn μn , yj n + ˜ − ˜ z Kn (μn ,yj n ,yj n ) Kn (μn ,yj n ,yj n )

. lim = cos π z. e n→∞ pn μn , yj n

(3.2)

We also proved [12, Theorem 2.2, p. 4] a result allows that varying .Qn based on asymptotics from [11]: Theorem 3.2 For .n ≥  1, let .In = (cn , dn ), where .−∞ ≤ cn < dn ≤ ∞. Assume that for some .r ∗ > 1, . −r ∗ , r ∗ ⊂ In , for all .n ≥ 1. Assume that μ'n (x) = e−2nQn (x) , x ∈ In ,

.

(3.3)

where (i) .Qn (x) / log (2 + |x|) has limit .∞ at .cn + and .dn − . (ii) .Q'n is strictly increasing and continuous in .In .   (iii) There exists .α ∈ (0, 1), .C > 0 such that for .n ≥ 1 and .x, y ∈ −r ∗ , r ∗ , .

  Q' (x) − Q' (y) ≤ C |x − y|α . n n

(3.4)

  (iv) There exists .α1 ∈ 12 , 1 , .C1 > 0, and an open neighborhood .I0 of 1 and .−1, such that for .n ≥ 1 and .x, y ∈ In ∩ I0 , .

  Q' (x) − Q' (y) ≤ C1 |x − y|α1 . n n

(3.5)

(v) .[−1, 1] is the support of the equilibrium distribution .ωQn for .Qn . Let .J ⊂ (−1, 1) be compact. Then uniformly for .yj n in .J, that is a local maximum of .|pn (μn , ·)| e−nQn (·) and for z in compact subsets of the plane, we have   z nQ'n (yj n ) pn μn , yj n + ˜ − ˜ z Kn (μn ,yj n ,yj n ) Kn (μn ,yj n ,yj n )

. lim = cos π z. (3.6) e n→∞ pn μn , yj n

On Local Asymptotics for Orthonormal Polynomials

185

We deduced Theorems 3.1 and 3.2 from a general proposition for a sequence of measures .{μn } [12, Theorem 2.3, p. 5]. It involves the sinc kernel S (t) =

.

sin π t . πt

Theorem 3.3 Assume that for .n ≥ 1 we have a measure .μn supported on the real line with infinitely many points in its support, and all finite power moments. Let .{ξn } be a bounded sequence of real numbers, and .{τn } be a sequence of positive numbers that is bounded above and below by positive constants, while .{Ψn } is a sequence of real numbers. Assume that uniformly for .a, b in compact subsets of .C,  Kn μn , ξn + .

lim

n→∞

aτn n , ξn

+

bτn n



Kn (μn , ξn , ξn )

eΨn (a+b) = S (a − b) .

(3.7)

Let us be given some infinite sequence of integers .T . The following are equivalent: (I)      n  τn n  1 1  1  < ∞ and sup . sup  + Ψ n

2 < ∞. n 2 ξ − x n n jn n∈ T n∈T   ξ − x n j n j =1 j =1 (3.8) (II) For each .R > 0, there exists .CR such that    p μ , ξ + τn z

   n n n n eΨn z  ≤ CR . . sup sup    p , ξ (μ ) n n n n∈T |z|≤R

(3.9)

(III) From every subsequence of .T , there is a further subsequence .S such that

pn μn , ξn + zτnn Ψn z α e = cos (π z) + sin π z, . lim π pn (μn , ξn ) n→∞,n∈S

(3.10)

uniformly for z in compact subsets of .C, where 

α=

.

τn pn' (μn , ξn ) + Ψn n→∞,n∈S n pn (μn , ξn ) lim

and .α is bounded independently of .S.

 (3.11)

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This equivalence is an exponential weight analogue of similar results in [13]. In [12], we also considered local limits at the soft edge. These involve the Airy function and kernel, defined as follows:  Ai(a)Ai ' (b)−Ai ' (a)Ai(b) , a /= b, a−b .Ai (a, b) = (3.12) 2 2 ' Ai (a) − aAi (a) , a = b. Ai is the Airy function, defined on the real line by Vallee and Soares [33] Ai (x) =

.

1 π





∞

cos 0

 1 3 t + xt dt. 3

(3.13)

The Airy function satisfies the differential equation Ai '' (z) − zAi (z) = 0.

.

(3.14)

We proved the following edge analogue of Theorem 3.3: Theorem 3.4 Assume that for .n ≥ 1 we have a measure .μn supported on the real line with infinitely many points in its support, and all finite power moments. Let .{ρn } be a sequence of positive numbers with limit 0, while .{Фn } is a sequence of real numbers, such that uniformly for .u, v in compact subsets of .C, .

Kn (1 + ρn u, 1 + ρn v) −Фn (u+v) Ai (u, v) . e = n→∞ Kn (1, 1) Ai (0, 0) lim

(3.15)

Let us be given some infinite sequence of integers .T . The following are equivalent: (I)      n n    1 1 2   + Фn  < ∞ and sup ρn . sup ρn

2 < ∞.  1 − x jn n∈T  n∈T  j =1 1 − xj n j =1 (3.16) (II) For each .R > 0, there exists .CR such that    pn (1 + ρn z) Ф z  n   . sup sup  p (1) e  ≤ CR . n n∈T |z|≤R

(3.17)

(III) From every subsequence of .T , there is a further subsequence .S such that .

  pn (1 + ρn z) Фn z Ai ' (z) e + c0 Ai (z) Ai ' (0) − Ai ' (z) Ai (0) , = ' pn (1) Ai (0) n→∞,n∈S (3.18) lim

On Local Asymptotics for Orthonormal Polynomials

187

uniformly for z in compact subsets of .C, where c0 =

.

  pn' (1) ρ + Ф n n pn (1) Ai ' (0)2 n→∞,n∈S 1

lim

(3.19)

and .c0 is bounded independently of .S. Unfortunately, the universality limit at the soft edge, namely (3.15), seems to require far more restrictive conditions on the weight than those inside the “bulk” of the support. The most general results have been established using deep RiemannHilbert techniques [8, 18, 19].

4 Fixed Exponential Weights As noted above, it was Freud who first began to consider quantitative features of orthogonal polynomials for exponential weights .W 2 = e−2Q . He usually considerd Q of polynomial growth at .∞, the prime example being .Q (x) = |x|α , .α > 0. Weights .e−2Q where Q grows faster than any polynomial, are often called Erd˝os weights, due to Erd˝os’ paper on them [6]. Subsequently there were efforts to investigate exponential weights on both finite and infinite intervals, and of Q of all rates of growth. Here is one such class of weights from [9, p. 7]: Definition 4.1 Let .I = (c, d) be an open interval, bounded or unbounded, containing 0. Let .W = e−Q , where .Q : I → [0, ∞) satisfies the following conditions: (a) .Q' is continuous in I and .Q (0) = 0. (b) .Q'' exists and is positive in .I \ {0}; (c) .

lim Q (t) = ∞.

|t|→∞

(d) The function T (t) =

.

tQ' (t) , t /= 0, Q (t)

is quasi-increasing in .(0, d), in the sense that for some .C > 0, 0 < x < y < d ⇒ T (x) ≤ CT (y) .

.

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E. Levin and D. S. Lubinsky

We assume, with an analogous definition, that T is quasi-decreasing in .(c, 0). In addition, we assume that for some .Λ > 1, T (t) ≥ Λ in I \ {0} .

.

(e) There exists .C1 > 0 such that .

Q' (x) Q'' (x) a.e. x ∈ I \ {0} . ≤ C 1 |Q' (x)| Q (x)



Then we write .W ∈ F C 2 . Examples of weights in this class are .W = exp (−Q), where  Q (x) =

.

Ax α , x ∈ [0, ∞) , B |x|β , x ∈ (−∞, 0)

where .α, β > 1 and .A, B > 0. More generally, if .expk = exp (exp (... exp ())) denotes the kth iterated exponential, we may take  Q (x) =

.

expk (Ax α ) − expk (0) , x ∈ [0, ∞)

exp𝓁 B |x|β − exp𝓁 (0) , x ∈ (−∞, 0)

where .k, 𝓁 ≥ 1, .α, β > 1. An example on .I = (−1, 1) is ⎧ 

−α  ⎨ exp 1 − x2 − expk (1) , x ∈ [0, 1) k   , .Q (x) =

−β ⎩ exp 1 − x2 − exp𝓁 (1) , x ∈ (−1, 0) 𝓁 where .α, β > 0. A key descriptive role is played by the Mhaskar-Rakhmanov-Saff numbers a−n < 0 < an ,

.

defined for .n ≥ 1 by the equations 1 .n = π 0=

1 π

 

an a−n an a−n

xQ' (x) dx; √ (x − a−n ) (an − x) Q' (x) dx. √ (x − a−n ) (an − x)

In the case where Q is even, .a−n = −an . The existence and uniqueness of these numbers is established in the monographs [9, 25], but goes back to earlier work of Mhaskar, Rakhmanov, and Saff [20, 21, 24].

On Local Asymptotics for Orthonormal Polynomials

189

We also define, βn =

.

1 1 (an + a−n ) and δn = (an + |a−n |) , 2 2

which are respectively the center, and half-length of the Mhaskar-Rakhmanov-Saff interval Δn = [a−n , an ] .

.

The linear transformation Ln (x) =

.

x − βn δn

maps .Δn onto .[−1, 1]. Its inverse Ln[−1] (u) = βn + uδn

.

maps .[−1, 1] onto .Δn . For .0 < ε < 1, we let Jn (ε) = Ln[−1] [−1 + ε, 1 − ε] = [a−n + εδn , an − εδn ] .

.



We let .pn W 2 , x denote the nth orthonormal polynomial for .W 2 , so that  .

    pn W 2 , x pm W 2 , x W 2 (x) dx = δmn .

I

Moreover, we let n−1        Kn W 2 , x, t = pj W 2 , x pj W 2 , t

.

j =0

and     K˜ n W 2 , x, t = W (x) W (t) Kn W 2 , x, t .

.

The new result of this paper is:



Theorem 4.2 Let .W = exp (−Q) ∈ F C 2 . Let .0 < ε < 1. Then uniformly for z in compact subsets of the plane, and uniformly for .ykn ∈ Jn (ε), we have  pn W 2 , ykn + .

lim

n→∞

 z K˜ n (W 2 ,ykn ,ykn )

pn (ykn )

e

−

Q' (ykn )z K˜ n W 2 ,ykn ,ykn

(

) = cos π z.

(4.1)

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In particular, if W is even, this holds uniformly for .|ykn | ≤ (1 − ε) an . To prove this result, we apply Theorem 3.3 with  1  [−1] Q Ln (x) , x ∈ Ln (I ) ; n

Qn (x) =

.

(4.2)

Wn (x) = exp (−Qn (x)) , x ∈ Ln (I ) ;

(4.3)

dμn (x) = Wn2n (x) dx.

(4.4)

.

.

Observe that with the definition of .Ln above Wn2n = W 2 ◦ Ln[−1] .

.

(4.5)

The orthogonal polynomials .pm (μn , x) are related to those for .W 2 by the identity   1/2 pm (μn , x) = δn pm W 2 , Ln[−1] (x) .

.

(4.6)

This  is 2easily

 established by a substitution in the orthonormality relations for pn W , x . The reproducing kernel .Kn (μn , x, t) for .Wn2n is related to the

reproducing kernel .Kn W 2 , x, t for .W 2 by the identity

.

  Kn (μn , x, t) = δn Kn W 2 , Ln[−1] (x) , Ln[−1] (t) .

.

(4.7)



In what follows, we shall denote the zeros of .pn W 2 , x by xnn < xn−1,n < ... < x2n < x1n ,

.

and the zeros of .pn (μn , x) by xˆnn < xˆn−1,n < ... < xˆ2n < xˆ1n .

.

From (4.6),

xˆj n = Ln xj n .

.

(4.8)



' We denote the zeros of . pn W 2 , x W (x) by .yj n , so that

yj n ∈ xj +1,n , xj n , 1 ≤ j ≤ n − 1.

.



' We denote the zeros of . pn (μn , x) e−nQn (x) by .yˆj n so that

yˆj n = Ln yj n ∈ xˆj +1,n , xˆj n .

.

(4.9)

On Local Asymptotics for Orthonormal Polynomials

191

Because of the linear nature of .Ln , we see that xˆj n − xˆkn =

xj n − xkn ; δn

(4.10)

xˆj n − yˆkn =

xj n − ykn . δn

(4.11)

.

.

We need some technical lemmas. For sequences .{cn }, .{dn } of non-0 real numbers, we write cn ∼ dn

.

if there exist positive constants .C1 , .C2 such that for .n ≥ 1, C1 ≤ cn /dn ≤ C2 .

.

Similar notation is used for sequences of functions. In addition, .C, C1 , C2 , ... denote constants independent of .n, x, t that may be different in different occurrences.

Lemma 4.3 Let .W ∈ F C 2 . Let .0 < ε < 1. There exists .n0 such that uniformly for .n ≥ n0 and .xˆj n ∈ [−1 + ε, 1 − ε], xˆj n − xˆj +1,n ∼

.

1 . n

(4.12)

Proof It is shown in [9, Corollary 13.4, p. 361] that uniformly for .n ≥ 1, 1 ≤ j ≤ n − 1,

xj n − xj +1,n ∼ ϕn xj n

(4.13)

.

where [9, p. 19] for .x ∈ [a−n , an ] , |x − a−2n | |x − a2n | ϕn (x) = √ . n [|x − a−n | + |a−n η−n |] [|x − an | + |an ηn |]

.

Moreover, .ϕn (x) = ϕn (an ) for .x > an and .ϕn (x) = ϕn (a−n ), for .x < a−n . Here [9, p. 15]  η±n = nT (a±n )

.



|a±n | δn

−2/3

= o (1)

(See [9, Lemma 3.7,  p. 76]for the .o (1) relation). The class of weights in Corollary

13.4 in [9] was .F lip 12 + . As noted in [9, p. 14], this class contains .F C 2 . We

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E. Levin and D. S. Lubinsky

now estimate .ϕn for .x ∈ Ln[−1] [−1 + ε, 1 − ε] = [−an + εδn , an − εδn ]. Since .a±2n ∼ a±n [9, Lemma 3.5(a), pp. 71–72] we see that for such x, δn δ2 ϕn (x) ∼ n = . n n δn2

(4.14)

.

Thus uniformly for .xj n ∈ Ln[−1] [−1 + ε, 1 − ε], (4.13) gives xj n − xj +1,n ∼

.

δn . n █

Then (4.12) follows from (4.10).

.

Next we need an estimate for the distance between critical points and zeros:

Lemma 4.4 Let .W ∈ F C 2 . Let .0 < ε < 1. There exists .n0 such that uniformly for .n ≥ n0 and .xˆj n ∈ [−1 + ε, 1 − ε], xˆj n − yˆj n ∼ yˆj n − xˆj +1,n ∼

.

1 . n

(4.15)

Proof We analyze the spacing between .xj n , xj +1,n and .yj n . Now .

 

pn W 2 , yj n W yj n  

 

= pn W 2 , yj n W yj n − pn W 2 , xj +1 ,n W xj +1,n  '   = pn W 2 , x W (x)

|x=ζ



yj n − xj +1,n ,

(4.16)

and a for some .ζ between .yj n and .xj +1,n . We find an upper bound for the derivative lower bound for the orthogonal polynomial at .yj n . Now .xj +1,n+1 ∈ xj +1,n , xj n ,  



and .pn W 2 , x W 2 (x) has its maximum in . xj +1,n , xj n at .yj n , so .

      pn W 2 , yj n W yj n    

  ≥ pn W 2 , xj +1,n+1 W xj +1,n+1   1/4 −1  xj +1,n+1 − an+1 (1 + ηn+1 ) xj +1,n+1 − a−n−1 (1 + η−n−1 ) ∼ δn+1 ,

On Local Asymptotics for Orthonormal Polynomials

193

by Theorem 13.2 in [9, Theorem 13.2, p. 360], uniformly in .1 ≤ j ≤ n. As above η±(n+1) = o (1) and for .xj n ∈ Ln[−1] [−1 + ε, 1 − ε] ,

.

.



 xj +1,n+1 − a±(n+1) 1 + η±(n+1)    ≥ C xj n − a±n  ≥ Cδn ,

so that uniformly for .xj n ∈ Ln[−1] [−1 + ε, 1 − ε] .

      −1/2 pn W 2 , yj n W yj n  ≥ Cδn .

(4.17)

To estimate the derivative, we use [9, p. 22, Theorem 1.17] .

      sup pn W 2 , x W (x) |(x − a−n ) (an − x)|1/4 ∼ 1.

(4.18)

x∈I

Let   (x − a ) (a − x) −n n , R (x) = pn W 2 , x δn2

.

a polynomial of degree .n + 2. From (4.18), we have for .x ∈ [a−n−2 , an+2 ], that .

|R (x)| W (x) ≤ C

|(x − a−n ) (an − x)|3/4 −1/2 ≤ Cδn . δn2

By a restricted range inequality in [9, Theorem 4.2(a), p. 96], .

−1/2

sup |R (x)| W (x) ≤ Cδn

.

x∈I

Then a Markov-Bernstein inequality in [9, Theorem 10.1, p. 293] yields .

  −1/2 sup (RW )' (x) ϕn+2 (x) ≤ C sup |R (x)| W (x) ≤ Cδn . x∈I

x∈I

Next, if .x ∈ Ln[−1] [−1 + ε, 1 − ε], this and (4.14) give    '  |(x − a ) (a − x)|  −n n 2   .  pn W , x W (x)  δn2        d [(x − a−n ) (an − x)]   dx −3/2 2 + Cnδn ≤  pn W , x W (x)  δn2

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≤ C |(x − a−n ) (an − x)|−1/4 −1/2

≤ Cδn

−3/2

|2x − βn | −3/2 + Cnδn δn2

−3/2

+ Cnδn

≤ Cnδn

,

by (4.18). Then we obtain for all .x ∈ L[−1] [−1 + ε, 1 − ε] , n  '    2  −3/2   .  pn W , x W (x)  ≤ Cnδn . In particular, this is is true for .x = ζ in (4.16), so that combining (4.16), (4.17), −3/2

−1/2

≤ Cnδn

Cδn

.

yj n − xj +1,n



so that yj n − xj +1,n ≥ C

.

δn . n

By (4.11), we can reformulate this as yˆj n − xˆj +1,n ≥

.

C . n

The corresponding upper bound follows from yˆj n − xˆj +1,n ≤ xˆj n − xˆj +1,n ≤

.

C , n

by the previous lemma. So we have shown yˆj n − xˆj +1,n ∼

.

1 . n

The proof for .xˆj n − yˆj n is similar.

█

.

By combining the two previous lemmas, we obtain the main estimates we need to apply Theorem 3.3:

Lemma 4.5 Let .W ∈ F C 2 . Let .0 < ε < 1. (a) There exists .n0 and C such that uniformly for .n ≥ n0 and .yˆkn [−1 + ε, 1 − ε], .

n 1  1

2 ≤ C. 2 n j =1 yˆkn − xˆ j n

∈

(4.19)

On Local Asymptotics for Orthonormal Polynomials

195

(b) n  .

j =1



1 − nQ'n yˆkn = 0. yˆkn − xˆj n

(4.20)

Proof (a) Let .0 < ε < ε' < 1. Lemma 4.4 shows that .

    yˆkn − xˆkn  , yˆkn − xˆk+1,n  ≥ C . n

This and Lemma 4.3 shows that there exists .C > 0 such that for .xˆj n , yˆkn ∈  −1 + ε' , 1 − ε' , .

  yˆkn − xˆj n  ≥ C 1 + |k − j | . n

Thus .

1 n2



1

xˆj n ∈[−1+ε' ,1−ε' ] yˆkn − xˆ j n

2 ≤ C

∞ 

1

(1 + |k − j |)2 k=−∞

≤ C.

The remaining terms may be estimated for .yˆkn ∈ [−1 + ε, 1 − ε] by .

1 n2



1 Cn

2 ≤ 2 ≤ C. n ˆkn − xˆj n ' ,1−ε ' ] y xˆj n ∈[−1+ε /

(b) We have 2

' pn W , x W (x) |x=y kn

.0 = pn W 2 , ykn W (ykn ) =

n  j =1

1 − Q' (ykn ) . ykn − xj n

Next from (4.2), Q'n (x) =

.

 1 '  [−1] Q Ln (x) δn n

(4.21)

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so that 1 Q'n yˆkn = Q' (ykn ) δn . n

.

Using (4.11), we reformulate (4.21) as n  .

j =1



nQ'n yˆkn 1

− = 0, δn δn yˆkn − xˆj n █

giving (4.20).

.

Finally, we need universality limits. Recall that .S (t) =

sin π t πt :

Lemma 4.6 Let .0 < ε < 1. Uniformly for .t ∈ [−1 + ε, 1 − ε], and .a, b in compact subsets of the plane,  Kn μn , t + .

lim

a ,t K˜ n (μn ,t,t)

+

b K˜ n (μn ,t,t)

 e

Kn (μn , t, t)

n→∞

−

nQ'n (t) (a+b) K˜ n (μn ,t,t)

= S (a − b) . (4.22)

Proof It was proved in [10, Theorem 7.4, p. 771] for the bigger class of weights F (dini), that uniformly for .x ∈ Jn (ε) = Ln[−1] [−1 + ε, 1 − ε], and .a, b in compact subsets of the real line,

.

 Kn W 2 , x + .

 a ,x K˜ n (W 2 ,x,x )

Kn W 2 , x, x

lim

n→∞

+

b K˜ n (W 2 ,x,x )

= S (a − b) .

This was actually deduced by applying Theorem 1.2 there. With .{μn } as above, Theorem 1.2 there first gave that for .t ∈ [−1 + ε, 1 − ε] ,  Kn μn , t + .

lim

n→∞

a ,t K˜ n (μn ,t,t)

+

b K˜ n (μn ,t,t)



Kn (μn , t, t)

= S (a − b) .

(4.23)

As noted at (1.13) in [10, p. 749, (1.13)], the proof of Theorem 1.2 showed that uniformly for .t ∈ [−1 + ε, 1 − ε], and .a, b in compact subsets of the complex plane,  Kn μn , t + .

lim

n→∞

a ,t K˜ n (μn ,t,t)

+

Kn (μn , t, t)

b K˜ n (μn ,t,t)

 e

−

nQ'n (t) (a+b) K˜ n (μn ,t,t)

= S (a − b) .

█

.

On Local Asymptotics for Orthonormal Polynomials

197

Proof of Theorem 4.2 We apply Theorem 3.3 to the measures .{μn }. We choose in Theorem 3.3, ξn = yˆkn ;

.

τn =

n

; K˜ n μn , yˆkn , yˆkn

nQ'n yˆkn Ψn = −

. K˜ n μn , yˆkn , yˆkn

First note that Lemma 4.6 gives the universality limit (3.7) in Theorem 3.3 with these choices of .ξn , τn , and .Ψn . Moreover, it follows from Lemma 7.7 in [10, pp. 775–776] that .{τn } is bounded above and below. Next from Lemma 4.5(a), we have the second condition (3.8) with .ξn replaced by .yˆkn and .xj n replaced by .xˆj n . Next, Lemma 4.5(b) gives 1

.



K˜ n μn , yˆkn , yˆkn

n  j =1



nQ'n yˆkn 1 −

= 0, yˆkn − xˆj n K˜ n μn , yˆkn , yˆkn

which gives a much stronger form of the first condition in (3.8). Also .α = 0 in (3.11), as follows from this last relation. Thus (3.10) of Theorem 3.3 gives   z pn μn , yˆkn + ˜ nQ'n (yˆkn ) Kn (μn ,yˆkn ,yˆkn ) − z

. lim e K˜ n (μn ,yˆkn ,yˆkn ) = cos (π z) , n→∞ pn μn , yˆkn uniformly for z in compact subsets of the plane. Using (4.6) , (4.7), and (4.2), this is easily reformulated as (4.1). .█

References 1. A.I. Aptekarev, Asymptotics of orthogonal polynomials in a neighborhood of the endpoints on the interval of orthogonality. Math. Sb. (Russia) 76, 35–50 (1993) 2. V.M. Badkov, The asymptotic behavior of orthogonal polynomials (Russian). Mat. Sb. (N.S.) 109(151)(1), 46–59 (1979) 3. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, vol. 3. Courant Institute Lecture Notes (New York University Pres, New York, 1999) 4. P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999) 5. B. Eichinger, M. Luki´c, B. Simanek, An approach to universality using Weyl m-functions, manuscript.

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6. P. Erd˝os, On the distribution of roots of orthogonal polynomials, in Proceedings of the Conference on Constructive Theory of Functions, ed. by G. Alexits et al. (Akademiai Koado, Budapest, 1972), pp. 145–150 7. G. Freud, Orthogonal Polynomials (Pergamon Press/Akademiai Kiado, Budapest, 1971) 8. T. Kriecherbauer, K.T.-R. McLaughlin, Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Not. 1999, 299–333 (1999) 9. E. Levin, D.S. Lubinsky, Orthogonal Polynomials for Exponential Weights (Springer, New York, 2001) 10. E. Levin, D.S. Lubinsky, Universality limits in the bulk for varying measures. Adv. Math. 219, 743–779 (2008) 11. E. Levin, D.S. Lubinsky, Bounds and Asymptotics for Orthogonal Polynomials for Varying Weights. Springer Briefs in Mathematics (Springer, Cham, 2018) 12. E. Levin, D.S. Lubinsky, Local limits for orthogonal polynomials for varying weights via universality. J. Approx. Theory 254, 105394 (2020) 13. D.S. Lubinsky, Local asymptotics for orthonormal polynomials in the interior of the support via universality. Proc. Am. Math. Soc. 147, 3877–3886 (2019) 14. D.S. Lubinsky, Pointwise asymptotics for orthonormal polynomials at the endpoints of the interval via universality. Int. Maths Res. Not. 2020, 961–982 (2020). https://doi.org/10.1093/ imrn/rny042 15. D.S. Lubinsky, Local asymptotics for orthonormal polynomials on the unit circle via universality. J. Anal. Math. 141, 285–304 (2020) 16. D.S. Lubinsky, Correction to Lemma 4.2(a) and 4.3(d). J. Anal. Math. 144, 397–400 (2021) 17. D.S. Lubinsky, On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle. Math. Sbornik. 213, 31–49 (2022) 18. K.T.-R. McLaughlin, P. Miller, The ∂¯ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights. Int. Math. Res. Pap., Article ID 48673, 1–78 (2006) 19. K.T.-R. McLaughlin, P. Miller, The ∂¯ steepest descent method for orthogonal polynomials on the real line with varying weights. Int. Math. Res. Not., Art. ID rnn 075, 1–66 (2008) 20. H.N. Mhaskar, Introduction to the Theory of Weighted Polynomial Approximation (World Scientific, Singapore, 1996) 21. H.N. Mhaskar, E.B. Saff, Where does the sup norm of a weighted polynomial live? Constr. Approx. 1, 71–91 (1985) 22. P. Nevai, Orthogonal Polynomials, Memoirs of the AMS no. 213 (1979) 23. P. Nevai, Geza Freud, orthogonal polynomials and Christoffel functions: a case study. J. Approx. Theory 48, 3–167 (1986) 24. E.A. Rakhmanov, On asymptotic properties of polynomials orthogonal on the real axis. Math. USSR. Sbornik 47, 155–193 (1984) 25. E.B. Saff, V. Totik, Logarithmic Potentials with External Fields (Springer, New York, 1997) 26. B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2 (American Mathematical Society, Providence, 2005) 27. B. Simon, Szegö’s Theorem and its Descendants (Princeton University Press, Princeton, 2011) 28. H. Stahl, V. Totik, General Orthogonal Polynomials (Cambridge University Press, Cambridge, 1992) 29. G. Szeg˝o, Orthogonal Polynomials. American Mathematical Society Colloquium Publications (American Mathematical Society, Providence, 1975) 30. V. Totik, Universality and fine zero spacing on general sets. Arkiv Mat. 47, 361–391 (2009) 31. V. Totik, Universality under Szeg˝o’s condition. Canadian Math. Bull. 59, 211–224 (2016) 32. V. Totik, Oscillatory behavior of orthogonal polynomials. Acta Math. Hungar. 160, 453–467 (2020) 33. O. Vallee, M. Soares, Airy Functions and Applications to Physics (World Scientific, Singapore, 2004)

New Trends in Geometric Function Theory Khalida Inayat Noor and Mohsan Raza

1 Introduction Geometric function theory is a classical branch of mathematics which deals with the geometrical behaviour of analytic functions, and Riemann, Cauchy, Weierstrass, Koebe [40, 45, 81] being the pioneers in this field. Riemann mapping theorem [17] gave us the permission to use the open unit disc E instead of any arbitrary domain .D ⊂ C with at least two boundary points. Koebe discovered that if analytic functions have an additional property of being univalent in .E, then in this case the conformality and the assertion of Riemann mapping theorem are confirmed. The set of functions f which are analyticand univalent  in .E, and satisfying the normalization conditions .f (0) = 0 = f ' (0) − 1 was denoted by the class .S and has been the fundamental component for this area. On making use of the geometry of the image domain, certain sub-classes of .S were defined, and among these, the most significant are the classes .S ∗ and .C which, respectively, consist of starlike and convex univalent functions. Thus it can be seen that geometric function theory establishes a beautiful relation between geometry and analysis. Nevanlinna and many other notable researchers studied these classes, see [5, 26, 81, 98]. Kaplan [37] introduced the class .K ⊂ S of close-to-convex functions and provided the geometrical characterization of the functions in this class. Apart from the geometry of the image domain, these functions have been discussed with respect to estimates and bounds for the coefficients of the series expansion. The first result was due to Bieberbach’s estimates for the second

K. I. Noor Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan M. Raza () Department of Mathematics, Government College University Faisalabad, Faisalabad, Pakistan © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_12

199

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K. I. Noor and M. Raza

coefficient of the function f in the class .S , see [26]. These fundamental classes have been generalized and extended in several directions and attracted the attention of a large number of researchers in this fascinating area. In these studies, the geometry of .f (E) plays an important role. It has several applications in engineering sciences, fluid dynamics, water resources and machine learning etc. Here, in this chapter, a survey together with most recent developments, we shall focus on highlighting both the theoretical and applicable aspects in this field.

2 Preliminaries and Basic Concepts The class .A of normalized analytic functions f defined in .E = {z : |z| < 1, z ∈ C} is given as f (z) = z +

∞ 

.

an zn .

(1)

n=2

The class .S ⊂ A consists of single-valued functions .f ∈ A if it does not take the same value twice in .E. In this case, we call f to be univalent in .E. If .f satisfies the condition .f ' (z0 ) /= 0, .z0 ∈ E, then we say f is locally univalent or conformal at .z0 . The well-known member of .S is the Koebe function k which is given by k (z) =

.

z (1 − z)

2

=

1 4



1+z 1−z

2 −

1 4

(2)

and maps .E conformally onto the w-plane cut from . −1 4 to .−∞ along the negative real axis. The function k plays the role of the extremal function for the class .S . The class .S is preserved under the transformations of conjugation, rotation, dilation, square root and disk automorphism, see [26]. Recently, the techniques of subordination and convolution (Hadamard product) are developed as main tools to study many challenging problems in the field of geometric function theory. We have included some details of these concepts.

2.1 Subordination In [39], Lindlof introduced the idea of subordination and then Littlewood [40] presented the term and developed it by finding the fundamental relations. Since then a unified and significant theory has been formulated. Currently, subordination occupies an important place and assumes a worthwhile role in complex analysis.

New Trends in Geometric Function Theory

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In subordination concept, Schwarz functions play the vital role. For basic details, we refer to [2, 45]. Definition 1 Let .ω be analytic in .E and be given by ω (z) =

∞ 

.

dn zn ,

n=2

with .ω (0) = 0, and .|ω (z)| < 1, for .z ∈ E. Then .ω is called a Schwarz function. The widely used results for these functions are given below, see [45].   (i) .|ω (z)| ≤ |z| and .ω' (0) ≤ 1.   2 (ii) .ω' (z) ≤ 1−|ω(z)| 2 . 1−|z|

Equality occurs in these two cases for the function .ω (z) = eiθ z, where .θ is real. Now, we give the definition of subordination. Definition 2 Let f and g be two analytic functions defined in .E. Then f is subordinated to g (written as .f (z) ≺ g (z) or .f ≺ g) if there exists a Schwarz function .ω such that f (z) = g (ω (z)) , z ∈ E.

.

In particular, if g is univalent, then .f (0) = g (0) and .f (E) ⊂ g (z). The idea of differential subordination was introduced by Miller and Mocanu, see [43]. The first and second order differential subordinations play a key role in the development of this theory.

2.2 Convolution (Hadamard Product) We first refer to [17, 26, 85] for basic details and definitions of the concept of convolution or Hadamard product. Definition 3 Let f and g belong to the class .A with f be given by (1) and .g(z) = ∞  z+ bn zn . Then the convolution .f ∗ g of f and g is defined as n=2

.

(f ∗ g) (z) = (g ∗ f ) (z) = z +

∞  n=2

an bn zn .

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K. I. Noor and M. Raza

It can be observed that z = f (z), (i) .f (z) ∗ 1−z (ii) .f (z) ∗ k(z) = f (z) ∗

z (1−z)2

= zf ' (z),

where k is the Koebe function given in (2). The following result involving convolution is being extensively used by several researchers in their studies about univalent functions, see [86]. Lemma 1 If f is convex, g is starlike in .E and .φ is any analytic function in .E with φ(0) = 1, then

.

.

(f ∗ φg) (E) ⊂ Coφ(E), (f ∗ φ) (E)

(3)

where .Co .φ(E) denotes the closed convex hull of .φ(E).

2.3 Caratheodory Functions and Related Classes Here we overview the class .P of Caratheodory functions of positive real part. Many geometric quantities which characterize various subclasses of .S are closely related to these functions. The function in class .P were first noticed and studied by Caratheodory [26] in 1907 and later on, throughout the years, a significant theory of these functions with several extensions has been created. We proceed to define the class .P and briefly outline some of its essential properties which will be required in our study. Definition 4 Let p be analytic in . E with .p(0) = 1 and satisfy the property Re {p(z)} > 0, .z ∈ E. Then p is said to belong to the class .P. It can be represented by the Taylor series as

.

p(z) = 1 +

∞ 

.

pn z n ,

z ∈ E.

n=1

The function .L0 (z) = 1+z 1−z , a Mobius transformation is an example which belongs to .P and it plays the extremal role for several problems related to this class. It is interesting to note that the class .S and class .P are connected by a sufficient condition for univalency, known as Noshiro-Warschawski theorem, see [26], as follows. Theorem 1 (Noshiro-Warschawski) If for some real .α, .f ∈ A , .eiα f (z) ∈ P, then .f ∈ S in .E.

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2.4 Some Basic Properties of the Class P (i) The class P is not contained in class S . A simple counter example of the function is p∗ (z) = 1 + zn . It confirms this claim since p∗ is not univalent for n ≥ 2. (ii) The class P is a convex set. (iii) Subordination representation of p ∈ P is given as p(z) ≺ 1+z 1−z , z ∈ E, or p(z) ≺ L0 (z). (iv) Using subordination, the class P(φ) is defined as p(z) ≺ φ(z), where φ is convex and p is analytic in E with p(0) = 1. (v) Herglotz representation of class P is given as: p ∈ P if and only if 1 .p(z) = 2π

2π

L0 (ze−it )dμ(t), z ∈ E,

0

see [26], where μ is non-decreasing real-valued function such that

2π

dμ(t) =

0

2π. (vi) Class P is preserved under several well-known operations.

2.5 Some Extensions of Class P The Caratheodory functions can be defined in different form in several ways. We here mention some of these as follows: (i) Class .P(ϱ) Let p be analytic in .E with .p(0) = 1. Then .p ∈ P (ϱ) , .0 ≤ ϱ ≤ 1, if and only if .Re .p(z) > ϱ. Obviously .P (0) ≡ P and also .P (ϱ) ⊂ P, see [26]. The function p is called a Caratheodory function of order .ϱ. We can express .p ∈ P (ϱ) as: p(z) = (1 − ϱ) p1 (z) + ϱ, p1 ∈ P.

.

(4)

(ii) The Class .Pm It is a natural type of generalization of class .P introduced in [78] and is given as follows: A function p, analytic in .E with .p(0) = 1, belongs to the class .Pm , m ≥ 2 if and only if 1 .p(z) = 2

π −π

1 + ze−it dμ(t), 1 + ze−it

(5)

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where .μ is a real valued function of bounded variation on .[−π, π] and satisfies the condition π

π |dμ(t)| ≤ m.

dμ(t) = 2,

.

−π

(6)

−π

It can be observed that .P2 = P. We can express .p ∈ Pm as  p(z) =

.

   m 1 m 1 p1 (z) − − p2 (z), m ≥ 2, + 2 4 2 4

(7)

where .p1 , p2 ∈ P. Remark 1 The class .Pm can further be extended as .Pm (h), .m ≥ 2. Here h is a convex univalent function. Then using (7) we say .p ∈ Pm (h) if and only if .pi ≺ h, .i = 1, 2 and p is expressed by (7). (iii) The Class .P [A, B] An analytic function p, with .p(0) = 1, belongs to the class .P [A, B] , .−1 ≤ B ≤ 1+Az A ≤ 1, if and only if .p(z) ≺ 1+Bz , .z ∈ E, see [31]. We note that (i) .P [1, −1] = P, (ii) .P [1 − 2ϱ, 1] = P (ϱ) , (iii) .P [A, B] ⊂ P. The geometric meaning of .p ∈ P [A, B] is that it maps .E onto the domain Ω [A, B] given by

.

    A−B 1 + Az  < , B = / 1 . Ω [A, B] = w : w − 1 + Bz  1 − B 2

.

The domain .Ω [A, B] represents an open circular disk with diameter end points 1+A 1−A and .D2 = 1+B D1 = 1−B and centered on the real axis with .0 < D1 < 1 < D2 . 

1−A It is known [52] that .P [A, B] is a convex set and .P [A, B] ⊂ P 1−B ⊂ P. A relationship between .P and .P [A, B] can be established as: .p ∈ P if and only if

.

.

(1 + A) p(z) + (1 − A) ∈ P [A, B] , z ∈ E. (1 + B) p(z) + (1 − B)

(8)

The class .P [A, B] can be used to define its extended version as the class Pm [A, B] , .m ≥ 2, by taking .p1 , p2 from .P [A, B] in relation (7).

.

Remark 2 If we take .A = 1, B = −q, .q ∈ (0, 1) , then we have the class 1+z with .p(E) ⊂ Ω [1, −q] = Ωq . P [1, −q] = Pq and in this case .p(z) ≺ 1−qz

.

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205

The domain .Ωq is given as     1 1  < . Ωq = w : w − 1−q 1−q

.

The class .Pq and the domain .Ωq will be used in the forthcoming section involving q-calculus. (iv) The Class .P [A, B; α] Recently, a natural generalization of the class .P [A, B] is being considered by many researchers by studying functions p such that .p(z) ≺ hα (z), where  hα (z) =

.

1 + Az 1 + Bz

α , α ∈ (0, 1] , −1 ≤ B ≤ A ≤ 1.

By a simple computation, it can be shown that .hα is convex and univalent in E to justify the subordination condition in formulating the class .P [A, B; α]. Calculations give us

.

  1 − |A|α−1 Re h'α (z) = Reα |A − B| > 0, for all z ∈ E. 1 + |B|α+1

.

Hence by Noshiro-Warschawski theorem, .hα is univalent. Also, by using definition of convexity, it can be shown that  Re

.

zh'α (z) h'α (z)

' 

 = Re

1 + (A − B) αz − ABz2 (1 + Az) (1 + Bz)

≥ 0,

since .t (r) = 1 − (A − B) αr − ABr 2 is decreasing in .(0, 1) and .t (0) = 1. Thus the class .P [A, B; α] is well defined and consequently can be generalized to the class .Pm [A, B; α] , .m ≥ 2 and .α ∈ (0, 1] , where .p1 , p2 ∈ P [A, B; α] in relation (7). The following special cases are worth noting: Pm [1, −1; 1] = Pm . P2 [1, −1; 1] = P. .P2 [1, −1; α] = Pα . α

1−A .Pm [A, B; α] ⊂ Pm (ϱα ) , . ϱα = , see [76]. 1−B α => |arg p (z)| < απ . Here, with .α = 1 , α ; .p ∈ P (e) .P2 [1, −1; α] = P 2 2 1 , we have .|arg p(z)| < π . The class .P 1 is related to the right-half of .p ∈ P

(a) (b) (c) (d)

. .

4

2

2

the lemniscate of Bernoulli (see [90]) enclosing the region       Ω1/2 = w : Rew > 0; w 2 − 1 < 1 .

.

1 has been also under discussion in [83]. The class .P 2

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(f) Recently, the class .Pc mapping .E onto a crescent-like domain was introduced in [82, 89] as follows: Let p be analytic in .E with .p (0) = 1 and satisfying the inequality .

   2  p (z) − 1 < 2 |p (z)| ,

z ∈ E.

Then .p ∈ Pc . This class is characterized by subordinating relation: If .p ∈ Pc , then  1 + z2 = qc (z) . (9) .p (z) ≺ z + It is shown in [89] that .qc is convex in the disc .|z| < the class .Pc if and only if √

√

2 2 .

1+Az is in The function . 1+Bz

1−A 1+A √ ≤ 2 + 1. ≤ 1+B 1−B √ 1+Az ∈ Pc if and only if . 2 − 2 ≤ B < 0 and . 1−Az ∈ Pc for .

1 In particular . 1+Bz √ 2 − 1. .0 < A ≤

2−1≤

Remark 3 The classes .Pq and .Pc can be generalized by using relation (7). (v) The Class .P (pk ) We now discuss the class .P (pk ) ⊂ P which is related with conic domain. It has been introduced and studied by several authors, for some details, see [4, 33– 36, 64, 65, 75]. Kanas and Wisnoiska [35, 36] defined the domains .Ωk , .k ≥ 0 as follows:   Ωk = u + iv : u2 > k 2 (u − 1)2 + k 2 v 2 .

.

(10)

For fixed .k ≥ 0, .Ωk represents the conic regions bounded, successively, by the imaginary axis (.k = 0), the right branch of a hyperbola .(0 < k < 1), a parabola .(k = 1) and an ellipse .(k > 1). Also, it was observed that, for no choice of parameter .k(k > 1), .Ωk reduces to a disc. For these conic domains the following functions .φi (z) , (i = 0, 1, 2, 3) play the role of extremal functions, mapping ' .E onto .Ωk such that .φi (0) = 1 and .φ (0) > 0. These functions are presented as i follows: ⎧ φ0 (z) ⎪ ⎪ ⎨ φ1 (z) .φi (z) = ⎪ φ (z) ⎪ ⎩ 2 φ3 (z)

for k = 0, for k = 1, for 0 < k < 1, for k > 1,

(11)

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where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ .

φ0 (z) =

1+z , 1−z

√ 2 1+ z φ1 (z) = 1 + π2 log 1−√z ,   √  2 2 2 arccos (k) arctan h z , 1 + 1−k 2 sinh π ⎞ ⎛

φ2 (z) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ φ3 (z) = 1 + ⎪ ⎪ ⎩

2 1−k 2

y(z) √ t

⎜ π √ sin ⎝ 2R(t)

(12)

⎟ 1 dx √ ⎠ + 1−k 2, 2 2 1−t x

1−x 2

0

! ' " √ π R (t) z−√ t and .y (z) = 1− , .t ∈ (0, 1) is chosen such that .k = cosh 4R(t) . Here .R (t) is tz Legendre’s complete elliptic integral of first kind, see [34, 35]. Now, we define the class .P (pk ) by using the concept of subordination as: Let p be analytic in .E with .p (0) = 1. Then .p ∈ P (pk ) if and only if .p (z) ≺ pk = φi (z) , (i = 0, 1, 2, 3), where .φi are given in (11) and (12). Remark 4 For .p ∈ P (pk ) , it can easily be verified that: 

k , k ≥ 0, (i) .P (pk ) ⊂ P k+1 (ii) .p (z) = hσ (z) , where .h ∈ P and .σ =

2 π

arctan k1 .

We note the class .P (pk ) can be generalized to the class .Pm (pk ) by using relation (7), see [60]. (vi) The Class .k − P[A, B] Noor and Malik [64] introduced a new geometrical structures of oval and petal type shape as image domains .Ωk [A, B] and defined the class .k−P[A, B], by combining both the concepts of Janowski (circular domains) and conic domains. Let p be a Caratheodory function in .E. Then p belongs to the class .k − P[A, B] if and only if p(z) ≺ pk (A, B, z) (k ≥ 0, −1 ≤ B < A ≤ 1),

.

where pk (A, B, z) =

.

(A + 1)pk (z) − (A − 1) , (B + 1)pk (z) − (B − 1)

(13)

and .pk is given by (11) and (12). For geometrical meaning of domain .Ωk [A, B], we refer to [64]. Before proceeding further, we list here some of the results widely used by serval researchers in studying different aspects of this area of geometric function theory.

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Lemma 2 ([43]) Let h be convex univalent in .E, h(0) = 1 and .Re{sh(z) + t > 0}, s, t ∈ C. If p is analytic in .E with .p(0) = 1, then

.

.

 p(z) +

zp' (z) sp(z) + t

≺ h(z), z ∈ E

implies .p(z) ≺ h(z) in .E. Lemma 3 ([27]) Let .p ∈ P, z ∈ E and .z = reiθ . Then

2π

.

|p(reiθ )|κ dθ < c(κ)

0

1 , (1 − r)κ−1

where .λ > 1 and .c(λ) is a constant depending only on .λ. Lemma 4 ([43]) Let .ψ be convex univalent in .E with .ψ(0) = 1. Suppose also that λ is analytic in .E with .Re{λ(z) ≥ 0, z ∈ E. If p is analytic in .E, .p(0) = 1, then

.

p(z) + λ(z)zp' (z) ≺ ψ(z) z ∈ E

.

implies that .p(z) ≺ ψ(z) in .E. Lemma 5 Let .h ∈ P. Then, for .z = reiθ , .0 ≤ θ1 < θ2 ≤ 2π, z ∈ E, we have   . max   h∈P

θ2 θ1

    2r zh' (z)  . }dθ  ≤ π − 2 arccos Re{ h(z) 1 − r2

This result is a special case of one proved in [53]. Lemma 6 ([43]) Let .u = u1 + iu2 and .v = v1 + iv2 and a complex function ψ : D ⊂ C × C → C with conditions be given as:

.

(i) .ψ(u, v) is continuous in domain D. (ii) .(1, 0) ∈ D and .ψ(1, 0) > 0, 2 (iii) .Re {ψ(iu2 , v1 )} ≤ 0 whenever .(iu2 , v1 ) ∈ D and .v1 ≤ −1 2 (1 + u2 ).   Let .p(z) = 1 + c1 z + c2 z2+ · · · be analytic in .E with . p(z), zp' (z) ∈ D for all ' .z ∈ E. If .Re ψp(z), zp (z) > 0, .z ∈ E, then .Re {p(z)} > 0 in .E. Lemma 7 ([66]) Let p be analytic in .E and .p(0) = 1. Then, for .α ≥ 0; .z ∈ E, p + δ1

.

implies .p ∈ Pm .

zp' ∈ Pm p

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3 Some Linear Operators Linear operators are of great importance in geometric function theory and have attracted the attention of a large number of researchers. Some of these are listed as below. (i) Carlson-Shaffer Operator Let .φ(a, c, z), c /= 0, −1, −2, · · · be the incomplete beta function related to the Gauss hypergeometric function .2 F1 given by φ(a, c; z) = z2 F1 (1, a; c; z) =

.

∞  (a)n n=0

(c)n

zn+1 , (z ∈ E),

where (a)0 = 1, (a)n =

.

Γ (a + n) = a(a + 1), ...(a + n − 1), n > 1. Γ (a)

Here .Γ denotes the well-known gamma function. The function .φ(a, c, z) has analytic continuation to the z-plane cut along the positive real line from 1 to .∞. z We note that .φ(a, 1; z) = (1−z) a and .φ(2, 1; z) is the Koebe function. Carlson and Shaffer [13] defined a convolution operator on .A involving the incomplete beta function as L (a, c)f (z) = (φ(a, c) ∗ f ) (z) ,

.

f ∈A.

If .a = 0, −1, −2, · · · , then .L (a, c)f is a polynomial. If .a /= 0, −1, −2, · · · , then application of the root test shows that the infinite series for .L (a, c)f has the same  1  n n radius of convergence as that of f because . lim  (a) (c)n  = 1. n→∞

Hence .L (a, c)f maps .A onto itself. We observe that .L (a, c)f is the identity and if .c /= 0, −1, −2, · · · , .L (a, c) has a continuous inverse .L (c, a). This characteristic of .L (a, c) has been effectively used by Noor in [56, 72] to study certain subclasses of .A involving the .L (a, c) operator. The .L (a, c) operator provides a convergent representation of differentiation and integration. If .g(z) = zf ' (z), then .g (z) = L (2, 1)f (z) and .f (z) = L (1, 2)g (z). For the application of this operator in geometric function theory, it is normally assumed that .a /= 0, −1, −2, · · · , and .c /= 0, −1, −2, · · · . Special Case of .L (a, c) By choosing particular values of a and c in .L (a, c), Ruscheweyh and Noor defined the convolution operators. These are listed as:

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K. I. Noor and M. Raza

(a) Ruscheweyh Derivative Operator Ruscheweyh derivatives of f of order n are denoted as L (n + 1, 1)f (z) = D n f (z) , n = 0, 1, 2 · · · .

.

Using convolution, .D n f can be expressed as z ∗ f (z) = fn (z) ∗ f (z), (1 − z)n+1

D n f (z) =

.

with .fn (z) =

n = 0, 1, 2 · · · . Also, .D n f can be expressed as

z , (1−z)n+1

(n) D n f (z) = z zn−1 f (z) , n = 0, 1, 2, · · ·

.

The following identity can easily be verified and is extensively used in several problems such as inclusion results between the classes # $' z D n f (z) f (z) = (n + 1) D n+1 f (z) − nD n f (z).

.

Replacing .n = λ, λ > −1, we can generalize this operator as D λ f (z) =

.

z ∗ f (z). (1 − z)λ+1

For more details, we refer to [6, 38, 63, 84, 85]. (b) Noor Integral Operator The operator .L (1, n + 1)f (z) = In f (z) is called Noor integral operator introduced in [67]. This operator is analogous to Ruscheweyh derivative operator and is defined as follows: z −1 by taking Let .fn (z) = (1−z) n+1 , we define .fn fn (z) ∗ fn−1 (z) =

.

z . (1 − z)2

Then the operator .In : A → A was introduced by the convolution relation as In (z) = fn−1 (z) ∗ f (z).

.

(14)

Using (14), the recursive relation, as given below, can be obtained z(In+1 f (z))' = (n + 1)In f (z) − nIn+1 f (z).

.

(15)

The relation (15) is quite useful in establishing several inclusion properties. This operator has been studied in the general form by replacing .n with .λ > −1. For some work on this operator, see [41, 57, 58, 67, 68, 93].

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Here, we recall Salagean derivative operator [88]. For .λ ≥ 0 and .l ∈ N0 = {0, 1, 2, ...}, the Salagean operator .Dλl : A −→ A is defined as Dλ0 f (z) = f (z)

.

Dλ1 f (z) = (1 − λ) f (z) + λzf ' (z) = Dλ f (z)

'  Dλl f (z) = (1 − λ) Dλl−1 f (z) + λz Dλl−1 f (z) = Dλ Dλl−1 f (z) , z ∈ E. Remark 5 Recently, using convolution, a new operator has been defined in [22] by combining Noor Integral operator .In and Salagean operator .Dλl [88]. Let. λ ≥ 0 and l .l ∈ N. Then the operator .DI λ,n : A −→ A is defined as l DIλ,n f (z) = (Dλl ∗ In )f (z) .

.

By simple computations, we can have the following relation

' l l l z DIλ,n+1 f (z) = (n + 1) DIλ,n f (z) − nDIλ,n+1 f (z) .

.

(16)

(ii) Srivastava-Owa-Ruscheweyh Operator The fractional derivative of order .α, 0 ≤ α < 1 is defined as follows (see [38, 73, 74, 97]) Dzα f (z) =

.

d 1 Γ (1 − α) dz



z 0

f (t) dt, 0 ≤ α < 1, z ∈ E. (z − t)α

Here the function f is analytic in a simply connected domain in the complex plane containing the origin, and the multiplicity of .(z − t)−α is removed by requiring 0 α .log(z − t) ∈ R whenever .(z − t) > 0. Clearly .Dz f (z) = f (z). Using .Dz , an α operator .ℑ : A −→ A is defined as ℑα f (z) = Γ (2 − α)zα Dzα f (z)

.

=z+

∞  Γ (1 + j )Γ (2 − α) j =2

Γ (1 + j − α)

= φ(2, 2 − α; z) ∗ f (z), where .φ(a, c; z) is incomplete beta function.

aj zj

0 ≤ α < 1, (z ∈ E),

(17)

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K. I. Noor and M. Raza

By using the operator .ℑα , the Srivastava-Owa-Ruscheweyh operator .Lαb,β : A −→ A is defined as: z − (1 − β)z2 z , ∗ ψβ (z)ℑα f (z), ψβ (z) = b 1−z (1 − z2 ) z ∗ φ(z, 2 − α; −z) ∗ ψβ (z) ∗ f (z) (b > 0, 0 ≤ β < 1). = (1 − z)b

Lαb,β f (z) =

.

(18) Some Special Cases It can easily be seen that (a) .α = 0, β = 0 gives us .Lαb,0 f (z) = (b) (c)

z (1−z)b

∗ f (z),

0 n .L n+1,0 f (z) = D f (z), α α 0 ' .L n+1,β f (z) = (1 − β)ℑ f (z) + β(ℑ f (z)) .

For the operator .L0n+1,β f , we refer to [3]. For a multiplier transformation, see [14]. μ

(iii) Integral Operator .Lλ μ Let .λ > −1 and .μ > 0. Then the operator .Lλ : A → A is defined as: μ

μ

Lλ f (z) = Cλ

.

μ zλ

0

z

t t λ−1 (1 − )μ−1 f (t)dt z ∞

Γ (λ + μ + 1)  Γ (λ + n) an zn , = z+ Γ (λ + 1) Γ (λ + μ + n)

(19)

n=2

where .Γ denotes the gamma function. Related to (19) the following identity holds. μ+1

zLλ

.

μ

μ+1

f (z) = (λ + μ + 1)Lλ − (λ + μ)Lλ

.

(20)

Relation (20) is useful in studying this operator. Special Cases By taking particular values of .λ and .μ, we obtain the following well-known integral operators.

z (a) .L10 = 0 f (t) t dt, Alexander operator. z 1 (b) .L1 f (z) = 2z 0 f (t)dt, Libra integral operator.

z λ+1 (c) .L1λ f (z) = zλ 0 t λ−1 f (t)dt, .(λ > −1), Bernardi integral operator.

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(iv) One Parameter Jung-Kim-Srivastava Operator The Jung-Kim-Srivastava operator .I σ : A −→ A is defined as: I σ f (z) =

.

2σ zΓ (σ )

=z+



∞  n=2

0

(

z

z (log )σ −1 f (t)dt, (σ real) t

2 σ ) an zn . n+1

(21)

For this operator, we refer to [32]. From (21), the following identity can easily be obtained z[I σ −1 f (z)]' = 2I σ f (z) − I σ +1 f (z).

.

(22)

4 Comprehensive Subclasses of Analytic Functions We first define certain important classes of analytic function in a comprehensive manner and then study these classes with respect to some analytical and geometrical aspects. Definition 5 Let .f, φ ∈ A , .(φ ∗ f /= 0). Then .F (z) = (φ ∗ f ) (z) is said to ' (z) belong to the class .Rm (h) if and only if . zF F (z) ∈ Pm (h), .m ≥ 2 and .Pm (h) is as defined in Remark 1. Special Cases z z . Then .F (z) = 1−z ∗ f (z) = f (z) and in this case .f ∈ (i) Let .φ (z) = 1−z Rm (h). 1+Az α (ii) By Taking .h(z) = ( 1+Bz ) , we have .F ∈ Rm [A, B; α]. 1+Az α ) . In this case .f ∈ (iii) Let .φ (z) = k (z) , the Koebe function and .h(z) = ( 1+Bz z ' Vm [A, B; α]. That is .F (z) = 1−z ∗ f (z) = zf (z) and it follows that .f ∈ Vm [A, B; α] ⇔ zf ' (z) ∈ Rm [A, B; α].

It can be noted that (a) .Vm [1, −1, 1] = Vm , the class of functions with bounded boundary rotation. Also .Vm [1 − 2ρ, −1, 1] = Vm (ρ), see [76]. (b) With .h(z) = pk (z), we obtain the classes .Rm (pk ) and .Vm (pk ). Definition 6 Let .F (z) = (φ ∗ f ) (z) and let, for .m ≥ 2, δ ≥ 0 such that  (zF ' (z))' zF ' (z) ∈ Pm (h), z ∈ E. .J (F ) = (1 − δ) +δ F ' (z) F (z) Then f is said to belong to the class .Mmδ (F, h).

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For some details, we refer [18, 48, 62, 66]. Special Cases We observe that

 z ∗ f , pk ] = Rm (pk ). (i) .Mm1 [ 1−z ! " 1+Az α z ∗ f, ( 1+Bz (ii) .Mm0 1−z ) = Vm [A, B; α]. Also .M21 (F, h) = S ∗ (h), M20 (F, h) = C (h). For .A = 1, B = −1, α = 1 and .k = 0, we refer to [11, 12]. Taking .h = pk gives us the class .k − U S T and .k − C V , see [65, 69, 71]. Here .S ∗ (h) and .C (h) are well-known classes of starlike and convex functions of Ma and Minda type functions. In particular, for 1+z ∗ .h (z) = 1−z , we obtain the classes .S and .C of starlike and convex functions respectively. Definition 7 Let .F (z) = (φ ∗ f ) (z) , G (z) = (φ1 ∗ g) (z). Then .F v [F, G; h] if and only if., Qm .

 (zF ' (z))' zF ' (z) (1 − v) ∈ Pm (h), + v G' (z) G' (z)

∈

m ≥ 2.

for .G ∈ Vm (h1 ). Choosing suitable values of the parameters, we obtain several new and known classes of analytic functions. See, for example, [15, 37, 50, 51, 55, 59]. Special Cases z . In this case .F (z) = f (z) and .G (z) = g (z). Then, with (i) Take .φ(z) = 1−z 1+z ∗ 0 ∗ ∗ .h(z) = 1−z , we have .Qm [F, G; h] = Cm and .C2 = C is the well-known class of quasi-convex functions, see [70]. z (ii) Take .φ(z) = 1−z . In this case .F (z) = f (z) and .G (z) = g (z). Then, with 1+z 1 .h(z) = , we have .Qm [F, G; h] = Tm , where .Tm consists of generalized 1−z close-to-convex functions and when .m = 2, T2 ≡ K , the well-known class of close-to-convex functions. For more details, we refer [37, 50, 54].

Definition 8 Let .F (z) = (φ ∗ f ) (z) be analytic in .E. Then F belongs to the class Bm,β (h), β > 0, if there exists a function .G = φ ∗ g ∈ Rm [h] such that

.

.

zF ' (z) ∈ Pm (h), z ∈ E. F 1−β (z) Gβ (z)

This class is a generalized version of the one called the class of Bazilevic functions. For details, we refer to [9, 49, 61].

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Definition 9 Let .α1 > 0, λ > 0 and h be convex in .E. Also, let .F = φ ∗ f ∈ A . Then .F ∈ Bm (λ, α1 , h) if and only if .

   F (z) α1 zF ' (z) (1 − λ) (F (z))α1 −1 ∈ Pm (h), z ∈ E. + z F (z)

5 Coefficient Results We now proceed to study these classes in several directions. The results presented here will provide insight of the concepts of bounded boundary and bounded radius rotations. Some geometric characterization of generalized close-to-convexity, coefficient inequalities, distortion results, integral preserving criteria and radius problems will be included. A brief survey on the Hankel determinant problem with some new results for our classes will also be presented. The use of q-calculus in the area of geometric function theory will also be included in somedetail. ∞ zn In Definition 7, we take .v = 1, φ1 = n=1 n , F (z) = z + ∞ n , h (z) = 1+z and .h(z) = ( 1+Az )α . Then we have the following A z 1 n=2 n 1−z 1+Bz coefficient result. '

(z) ∈ Pm [A, B; α], g ∈ V2 ≡ C . Then Theorem 2 Let . zFg(z)

|An | ≤

.

{mα|A − B|(n − 1) + 4} . 4n

(23)

The proof involves routine calculations where we have used the coefficient estimates for .p ∈ Pm [A, B; α] and .g ∈ C . Remark 6 For .m = 2, g ∈ V2 , the function F is univalent and this implies F (z) F1 (z) = ww00−F (z) , .w0 /= 0, F (z) /= w0 is also univalent in .E.

.

Since .F1 (z) = z + (A2 + .

1 2 w0 )z

+ · · · , we have

1 1 | ≤ 2. − |A2 | ≤ |A2 + w0 w0

By using de Branges theorem for the bound of second coefficient of univalent  zF ' n function, see [16], we have for .F (z) = z + ∞ n=2 An z and . g ∈ P[A, B; α], .F (E) contains the Schlicht disc d such that  d = w : |w|
2 + 2−α 1−ρ , .ρ given by (24),

  

m −1 +α , An = O(1)nβ , β = (1 − ρ) 2

.

where .O(1) is a constant depending only on .α, m and .ρ. Proof Since F ' (z) = G' (z) p (z) ,

.

where .p ∈ Pα and .G ∈ Vm [A, B; α] ⊂ Vm (ρ) . Then using the result due to [76] and Cauchy theorem with .z = reiθ , we have 1 .n |An | ≤ 2π r n

2π

m |s (z)|1−ρ |p (z)|( 2 −1)(1−ρ) |p1 (z)|α dθ,

0

where .p, p1 ∈ P and .s ∈ S ∗ . Now by using distortion result for starlike functions, Lemma 3 and Holder’s inequality, we obtain the required result. Remark 7 Rephrasing Definition 7 with some particular values of parameters, we have the following classes: (i) For .φ (z) = φ1 (z) = 0 if and only if

z 1−z ,

λ1 α h(z) = ( 1+z 1−z ) , we say that .f ∈ Q2 [f, g, h], λ1 ≥

   1+z α (zf ' (z))' f ' (z) + λ1 ' . (1 − λ1 ) , g ∈ C. ≺ g ' (z) g (z) 1−z (ii) For .λ1 > 0, we can easily deduce that, for .f ∈ Q2λ1 [f, g, h] there exists a function .f1 ∈ Q20 [f, g, h] if and only if 1 1− λ1 .f (z) = z 1 λ1



z

1

ξ λ1 0

−2

f1 (ξ )dξ.

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217

6 Necessary Conditions z In Definition 7, by choosing .δ = 0, .φ(z) = 1−z and .F = φ ∗ f , we have the α

1+Az class .Vm (h). Let .h(z) = 1+Bz . Then .Vm (h) reduces to the class .Vm [A, B; α] which is a subclass of .Vm of functions with bounded boundary rotation. On taking .h(z) = pk (z), we have the class .Vm (pk ). Also in Definition 7, by choosing .v = 1, .φ∗f = F , .φ∗g = G, .G ∈ Vm [A, B; α] or .G ∈ Vm (pk ), we have the corresponding classes .Tm [A, B; α; h], .Tm [pk , h]. Here we shall discuss necessary conditions for these classes. Special cases of our results will yield several known and new ones, and hopefully motivate future research in this direction. It can easily be established that:

(i) Vm [A, B; α] ⊂ Vm (ρ1 ), ρ1 = .

and (ii) Vm (pk ) ⊂ Vm (ρ2 ), ρ2 =

1−A 1−B

α (25)

k k+1 .

Thus, we shall consider the classes .Tm (ρi , h) for .i = 1, 2 for necessary conditions. Goodman [25] defined the class .K (β) of analytic functions f , which are locally univalent for .z = reiθ , 0 ≤ θ1 < θ2 ≤ 2π, β ≥ 0 as follows: Definition 10 A function f of the form (1) belongs to .K (β) if it is analytic in .E, f ' (z) /= 0 and

.





θ2

Re

.

θ1

(zf ' (z))' dθ > −βπ, f ' (z)

β ≥ 0.

(26)

We interpret (26) with some geometrical meaning for .f ∈ K (β). For simplicity, let us suppose that the image domain is bounded by an analytic curve C. At a point on C, the outward drawn normal has an angle .arg{eiθ f ' (eiθ )}. Then it follows that the angle of the outward drawn normal turns back at most .βπ . This is a necessary condition for a function to belong to the class .K (β). When .β = 1, then f satisfying (26) is close-to-convex and therefore univalent in .E and therefore .K (1) coincides with the class .K of close-to-convex functions introduced by Kaplan in [37]. ' ' For locally univalent function .f ∈ Vm with . (zff ' ) ∈ Pm , it has been proved in [11] that

θ2

.

θ1



m  (zf ' (z))' dθ > − − 1 π, Re f ' (z) 2 

for .m ≥ 2, z = reiθ ; 0 ≤ θ1 < θ2 ≤ 2π .

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K. I. Noor and M. Raza

It can easily be seen that .f ∈ Vm is univalent (close-to-convex) for .2 ≤ m ≤ 4. Following result is a generalized version of one given in [59]. Theorem 4 Let .F = φ ∗ f and let .F ' (z) /= 0 in .E. Then .F ∈ Vm (ρi ), i = 1, 2 implies

θ2

.

θ1

! m " (zF ' (z))' − 1)(1 − ρ dθ > − ( ) π, Re i 2 F ' (z) 

(27)

where .m ≥ 2, z = r iθ , 0 ≤ θ1 < θ2 ≤ 2π and  ρ1 =

.

1−A 1−B

α , ρ2 =

k . k+1

(28)

Using Theorem 4 and Lemma 6, we have: '

F Theorem 5 Let .F = φ ∗ f and let F be locally univalent. Let . G ' ∈ P[1, −1; α] for .G ∈ Vm (ρi ), (G = φ ∗ g) where .ρi , .i = 1, 2 is given by (28). Then, for iθ .z = re , 0 ≤ θ1 < θ2 ≤ 2π,



θ2

.

θ1

! m " (zF ' (z))' dθ > − ( − 1)(1 − ρ Re ) + α . i F ' (z) 2 

(29)

The class of functions F satisfying the condition given in Theorem 5 is denoted by .Tm [F, ρi , α]. The proof follows immediately when we note that .F ' (z) = [G1 (z)]1−ρi hα1 (z) , G1 ∈ Vm , h1 ∈ P. Remark 8 (i) From (29), it follows that for .f ∈ Tm [F, ρi ; α], F = φ ∗ f is univalent in .E i) for .m < 2(2−α−ρ . 1−ρi # $ (ii) .Tm [F, ρi , α] ⊂ K ( m2 − 1)(1 − ρi ) + α , and the functions in .Tm [φ ∗ f, ρi , α] need not be finitely-valent. From the Remark 8(ii) and the results given in [25] for the class .K (β), we have the following result.  n Theorem 6 Let .f ∈ Tm [F, ρi , α] with .F = φ ∗ f and .F (z) = z + ∞ n=2 An z ; (i) Denote by .L(r, F ) the length of the image of the circle .|z| = r under .F . Then, for .0 ≤ r < 1, ∗ ∗ .L(r, F ) ≤ L(r, F ), where .F is defined by 1 .F (z) = b1 ∗

%

1+z 1−z

b2

& −1 =z+

∞  n=2

Dn z n ,

(30)

New Trends in Geometric Function Theory

219

with .

b1 = m(1 − ρi ) + 2(α + ρi ),   b2 = m2 − 1 (1 − ρi ) + (1 + α)

(31)

(ii) .|An | ≤ Dn , for .n = 2, 3, 4, ... where .Dn are defined by (30) and .b2 is an even integer. This result is sharp for each .n ≥ 2. (iii) For .z = reiθ , 0 ≤ r < 1. ' ( b b −1

1+r 2 1+r 2 1 ' −1 , , (b) .|F (z) ≤ b1 (a) .|F (z) ≤ 1−r 1−r where .b1 and .b2 are given by (31). These bounds are sharp, equality being for the function .F ∗ defined by (30) and (31). It can easily be shown, with routine calculations, that .F ∗ ∈ Tm (F, ρi , α). For this result, we refer to [59]. Remark 9 From Theorem 5 and a result due to [79]. We can easily deduce that Tm (F, ρi , α) is a linearly invariant family of order .b2 , where .b2 is given by (31).

.

From this observation, we have the following covering result. Theorem 7 The image of .E under the functions F in the class .Tm (F, ρi , α) contains the schlicht disc .|z| < 2b12 where .b2 is given by (31). α

1+Az or .h(z) = pk (z). Now, in Definition 6, if we take .F = φ∗f, and .h(z) = 1+Bz Then it follows that .F ∈ Mmδ (F, h) if and only if .

 (zF ' (z))' zF ' (z) (1 − δ) ∈ Pm (h) ⊂ Pm (ρi ), i = 1, 2, + δ F ' (z) F (z)

where  ρ1 =

.

1−A 1−B

α , ρ2 =

k (k ≥ 0), δ ≥ 0. k+1

Here we discuss a necessary condition for the functions of this generalized type of ' (z) class related with Mocanu function. Let . zF F (z) = p(z). Then p is analytic in .E, .p(0) = 1 and we have .

(zF ' (z))' zp' (z) , = p (z) + ' F (z) p (z)

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K. I. Noor and M. Raza

and so (1 − δ)

.

zp' (z) zF ' (z) (zF ' (z))' = p + (1 − δ) . + δ (z) F (z) p (z) F ' (z)

(32)

We take .δ1 = (1 − δ). This gives us p + δ1

.

zp' ∈ Pm (ρi ) p

Hence, for .z = reiθ , F ∈ Mmδ (F, h), we have

θ2

.

θ1

  θ2 (zF ' (z))' zp' (z) (zF ' (z)) dθ +δ dθ = Re p (z) + δ1 Re (1 − δ) F ' (z) p (z) F (z) θ1 m > −(1 − ρi ) π, (33) 2

where we have used the definition of .p ∈ Pm (ρi ). From (32) and (33) it easily follows that  θ2 zp' (z) m dθ > −(1 − ρi ) π, . Re p (z) + δ1 (34) 2 p (z) θ1 We note that, for . m2 (1 − ρi ) ≤ 1, we have from (34),

θ2

.

θ1

zp' (z) dθ > −π. Re p(z) + δ1 p(z) 

(35)

Using Lemma 7, we have

θ2

.

θ1

Re {p(z)} dθ > −π, for m ≤

2 . 1 − ρi

(36)

We can prove now: z and .F = φ ∗ f = f and let, for .δ ≥ 0, .f ∈ Mmδ (f, h) Theorem 8 Let .φ(z) = 1−z and .h ∈ Pm (ρi ), .ρi are given by (28). Then f is close-to-convex (univalent) in .E 2 for .m ≤ 1−ρ . i

By assigning different permissible values to the parameters .α, A and B or k, we obtain several special cases of this result.

New Trends in Geometric Function Theory

221

7 Some Inclusion and Radius Problems Now by using Noor-Salageen operator .DIλ,n given in Remark 5 for .l = 1, we define the related classes as follows: Let .F = φ ∗ f . Then F is said to belong to the class .Rm (λ, n; h) if and only if .DIλ,n (F ) ∈ Rm (h), .m ≥ 2, .λ, n ∈ N. We prove now the following theorem. Theorem 9 R2 (λ, n, h) ⊂ R2 (λ, n + 1, h), λ, n ∈ N0 , h ∈ P(h).

.

Proof Let .F ∈ R2 (λ, n, h) and suppose that .

z(DIλ,n+1 F (z))' = p(z). DIλ,n+1 F (z)

(37)

It can be seen that p is analytic in .E and .p(0) = 1. Logarithmic differentiation of (37) and using (16) with .l = 1 together with some computations, we get  zp' (z) z(DIλ,n F (z))' ≺ h(z). = p(z) + . p(z) + n DIλ,n F (z) Now applying Lemma 2, it follows .p(z) ≺ h(z) and the proof is complete. As a special case, we can consider .h ∈ P(ρi ), i = 1, 2 where .ρ1 , ρ2 are given by (28). Corollary 1 R2 (λ, n, ρi ) ⊂ R2 (λ, n, βi ), i = 1, 2,

.

where βi =

.

2(2nρi + 1)  , (2n − 2ρi + 1) + (2n − 2ρi + 1)2 + 8(2nρi + 1)

In case of .ρi = 0, then .βi =

√2

(2n+1)+

(2n+1)2 +8

i = 1, 2.

(38)

.

Proof We proceed by considering .p(z) = (1 − β)h(z) + β in (37) and with some computations, we obtain  Re (1 − β)h(z) + (β − ρi ) +

.

(1 − β)zh' (z) (n + β) + (1 − β)h(z)

> 0, (z ∈ E).

Now, by using well-known distortion and growth results for the .P(ρi ) class and frequently used techniques, we can easily prove this result.

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K. I. Noor and M. Raza

Choosing various permissible value of the parameters involved, we obtain several new and known results as special cases. For example, for .n = 0, ρi = 0, we have 1 .βi = . 2 Remark 10 (i) Similar inclusion results can be proved involving this linear operator for other generalized classes. (ii) With similar procedure and using differential subordination technique, it can be shown that these classes are preserved under the generalized Bernardi operator given by (19). 1 [F, G; h] = T and define (iii) In Definition 7, we consider the special case as .Qm m this class as: F' Let .F = φ ∗ f, G = φ ∗ g. Then . G ' ∈ Pm (h), G ∈ C (h1 ). That is .F ∈ ' Tm (F, G; h) if and only if . zF ∈ P m (h), for some .G1 ∈ R2 (h). Thus we G1 introduce the class .Tm (λ, n, h) for .λ, n ∈ N0 , h ∈ Pm (h) as follows: .F ∈ Tm (λ, n, h) if and only if .DIλ,n (F ) ∈ Tm (h). Inclusion result can be proved for the class .T2 (λ, n, h) as T2 (λ, n, h) ⊂ T2 (λ, n + 1, h)

.

Next, we discuss the following radius problem. '

' '

(zG ) F ∈ Theorem 10 Let .F = φ ∗ f, G = φ ∗ g and let . G ' ∈ P[1, −1, α] with . G' Pm (h). Then F is a convex function of order .ρi for .|z| < ri , where

ri =

.

2 2α ) , m1 = m + , m > 2, 1 − ρi m1 + m21 − 4

and .ρi are given by (28). Proof Since .Pm (h) ⊂ Pm (ρi ) and so .G ∈ Vm (ρi ). From the given conditions we can write F ' (z) = (G'1 (z))1−ρi (p(z))α , G1 ∈ Vm , p ∈ P.

.

(39)

Now, it is known [12] that for .G1 ∈ Vm , there exists a starlike function s and p2 ∈ P such that

.

m

zg2' (z) = s(z) (h1 (z)) 2 −1 ,

.

z ∈ E.

Now differentiating logarithmically, we obtain from (39) that 1+

.



m  zp' (z) zp' (z) zF '' (z) 2 +α + (1 − ρ ) p (z) + − 1 , = ρ i i 1 ' 2 p2 (z) p(z) F (z)

New Trends in Geometric Function Theory

where .p1 (z) =

zs ' (z) s(z) ,



(zF ' (z))' .Re − ρi F ' (z)

223

s ∈ S ∗ . That is,

 '     zp' (z)    (z)  ≥ (1 − ρi ) Re(p1 (z)) − m2 − 1  p22(z)  − α  zp p(z)  ≥ (1 − ρi )

!

1−r 1+r

−

 2r " − 1 1−r 2 −

m 2

2αr , 1−r 2

and this gives us 1 Re . 1 − ρi



(zF ' (z))' F ' (z)

≥

1− m+

2α 1−ρi 1 − r2



r + r2

=

T (r) . 1 − r2



2α 2α < + 1 = 2 − m + We note .T (0) = 1 > 0 and .T (1) = 1 − m − 1−ρ 1−ρi i 0. Therefore, .ri ∈ (0, 1) . Solving .T (r) = 0 gives us the required result. This completes the proof. Corollary 2 When .A = 1, B = −1, .ρi = 0, α = 1 and .G ∈ Vm . This gives us 2 √ the radius of convexity for .F ∈ Tm as .r0 = . Furthermore, the case 2 (m+2)+

m +4m

m = 2 can be settled separately and we have .r0 = 1√ . This is well known radius 2+ 3 of convexity for the class .K of close-to-convex functions.

.

By assigning other permissible values to the parameters .α, A, B and .m, we obtain several new and known results. Remark 11 Let .fn (z) = for .|z| < rn , where

z , (1−z)n+1

rn =

.

n ∈ N0 . Then it can be verified that .fn is convex

(3n + 1) +



2 (3n + 1)2 − 4n2

(40)

.

Theorem 11 Let .F = fn ∗ f and let .f ∈ T2 (g, h) with .g ∈ C (h) in .E. Then F ∈ T2 (G, h) in .|z| < rn , where .rn is given by (40) and .G = fn ∗ g.

.

Proof Proof is immediate since .G ∈ C (h) for .|z| < rn and from

 ' fn ∗ fg ' zg ' (z) (fn ∗zg ' )(z)

.

F ' (z) G' (z)

=

, we obtain the result by using Lemma 1.

It is known that .S ∗ (n+1, h) ⊂ S ∗ (n, h) ⊂ · · · S ∗ . We can show the following result. Theorem 12 Let .f ∈ S ∗ (h). Then .f ∈ S ∗ (n + 1, h) for .|z| < rn where .rn is given by (40).

224

K. I. Noor and M. Raza

Proof Consider '

fn ∗ zff f z(D n f )' fn ∗ hf . . = = fn ∗ f Dnf fn ∗ f Now .fn is convex in .|z| < rn and f is starlike. We apply Lemma 1 and have D n f ∈ S ∗ (h) for .|z| < rn . That implies .f ∈ S ∗ (n + 1, h) for .|z| < rn , where .rn is given by (40).

.

Theorem 13 Let Fλ =

.

λ+1 zλ



z

t λ−1 f (t)dt, λ > −1

(41)

0

and let .Fλ ∈ S ∗ (n, h). Then .f ∈ S ∗ (n, h) for .|z| < rλ∗ , .rλ∗ is given by (42). Proof From (41), we have f (z) =

.

where .pλ (z) = ∗ .|z| < r , where λ

∞

j +λ j j =1 1+λ z

λFλ + zFλ' = Fλ (z) ∗ pλ (z), λ+1 and it is known [10] that .pλ is convex in the disk 

∗ .rλ

=

√

2− 1 2,

3+λ2 , 1−λ

λ /= 1, λ = 1.

(42)

Thus .f ∈ S ∗ (n, h) in the disk .|z| < rλ∗ and this radius is best possible.

8 q-Calculus Approach to Geometric Function Theory ν [F, g; h] by In this section, we discuss here the special cases of the class .Qm 1+z z considering .ν = 0, 1, m = 2, .φ(z) = 1−z and .h(z) = 1−z . This will lead us to the classes .K and .C ∗ of close-to-convex and quasi-convex functions respectively. We shall generalize these classes by using q-calculus see [1, 34]. The q-calculus or quantum calculus is ordinary calculus without notion of limit. The formal power series were introduced by Christoph Guderman (1798–1852) and Karl Weirstrass (1815–1897). Recently it has attracted the serious attention of many researchers. It has several applications in different mathematical areas such as number theory, combinatorics, basic hypergeometric functions, orthogonal polynomials and other branches of science such as quantum theory, mechanics and theory of relativity. For details, we refer to [20, 21].

New Trends in Geometric Function Theory

225

The application of q-calculus was introduced by Jackson [29, 30] by introducing the q-difference operator and q-integral in a systematic way. Making use of qderivative, Ismail et al. [28] introduced the concept of q-starlike and q-convex functions. The q-analogue of close-to-convexity is defined in [87]. For more work in geometric function theory related to q-calculus, we refer to [91, 92, 94–96]. We first recall some basic definitions of q-calculus as follows: Let .f ∈ A and be given by (1). Then q-derivative of f , for .0 < q < 1, is defined by ∂q f (z) =

.

f (qz) − f (z) , (q − 1)z

z /= 0,

(43)

also .∂q f (0) = f ' (0) and .∂q2 f (z) = ∂q (∂q f (z)). Using (1), we deduce that ∂q f (z) = 1 +

∞ 

.

[n]q an zn−1 ,

n=2 n

where .[n]q = 1−q 1−q . When .q → 1−, [n]q → n. In 1990, Ismail et al. [28] generalized the class .Sq∗ of q-starlike functions by replacing the ordinary derivative with q-derivative and replacing the right half plane with the domain .|w − (1 − q)−1 | ≤ (1 − q)−1 . Thus we have the following: ∗ .Sq

    z(∂q f )(z) 1 1   ≤ , 0 < q < 1, z ∈ E , − = f ∈A : 1−q 1−q f (z)

and     ∂q (z∂q f )(z) 1 1   .Cq = f ∈ A :  ∂ f (z) − 1 − q  ≤ 1 − q , 0 < q < 1, z ∈ E . q In [87], the q-analogue of close-to-convex functions is defined as: Kq =

.

⎧ ⎨ ⎩

F ∈A ;G∈

* 0 0 .

V →∞

(2.32)

Since .ηΛ is given by (2.17), the both .σ and .ηΛ (σ ) are three-component vectors, and .(ηΛ (σ ))2 = ‖ηΛ (σ )‖2 . Hence, the space-averaged magnetization: .η (2.27), fluctuates in the state .ωβ . The following well-known proposition relates ODLRO and SSB for the Heisenberg ferromagnet: Proposition 1 If .ωβ exhibits conventional ODLRO, it undergoes the SSB defined by (2.20), with A defined by (2.27). Conversely, if (2.20) holds for some .ωβ,n in the decomposition (2.29), with A given by (2.27), then (2.32) holds. Proof If .ωβ exhibits ODLRO, it follows from (2.21) and (2.28) that .λ /= 0 in (2.28), and thus the ergodic decomposition (2.29) is nontrivial. hence, SSB holds, with A in (2.20) given by .η(A), defined by (2.27). The converse statement is a direct consequence of the ergodicity of .ωβ,n , and the fact that (2.20) implies that .λ /= 0. ⨆ ⨅ Remark 1 The ergodic states are not invariant under .G = SO(3) but rather under the isotropy (stationary) subgroup .Hn0 of G, and .Sd−1 may be identified as the harmonic space .G/H . Remark 2 The connection between ODLRO and the existence of several equilibrium states for quantum spin systems was first pointed out by Dyson, Lieb and Simon in their seminal paper [22], see also the review by Nachtergaele [27] and references given there. By Lieb et al. [22], both the spin one-half XY model for 1 2 .β ≥ βc , and the Heisenberg antiferromagnet for suitable spin and .β ≥ βc , with 1 2 .βc , βc explicitly given in [22], display ODLRO in the sense of Definition 2, but with different .ηΛ in (2.27) (for the antiferromagnet the sum over .x ∈ Λ being replaced by a sum over .Λ∩A, where .Zd = A∪B, A and B being disjoint sublattices. We expect that Proposition 2 is applicable to the above mentioned cases, yielding SSB (of the rotation group (.SO(2) in the XY case) according to Definition 2, but a choice of .ηΛ for a general Heisenberg hamiltonian, with arbitrary spin, is not known, as well as what the correct order parameters are, and how the set of pure phases should be parametrized and constructed (We thank B. Nachtergaele for this last remark). We therefore restrict ourselves to the ferromagnet as our quantum spin example. Remark 3 By (2.28) we have different values for the “charge density” .η(σ ) labelled by .n ∈ S2 . By a well-known result (see, e.g., [12], Corollary 6.3), the GNS representations .πωn associated to the corresponding states .ωn in the (central) decomposition (2.30) are not unitary equivalent (they are, more precisely, disjoint, see Definition 6.6 in [12]), and the GNS Hilbert space splits into a direct integral of disjoint “sectors” .Hn (see e.g. [10]). We note that in this respect the case of boson systems is more complicated than spin lattice systems. It becomes clear even on the level of the perfect Bose-gas

440

W. F. Wreszinski and V. A. Zagrebnov

(PBG). To see this, consider PBG in a three-dimensional anisotropic parallelepiped Λ := V α1 ×V α2 ×V α3 , with periodic boundary condition (p.b.c.) and .α1 ≥ α2 ≥ α3 , .α1 + α2 + α3 = 1, i.e. the volume .|Λ| = V . In the boson Fock space .FΛ := Fsymm (L2 (Λ)) the Hamiltonian of this system for the grand-canonical ensemble with chemical potential .μ < 0 is defined by self-adjoint operator .

H0,Λ,μ := TΛ − μ NΛ =



(εk − μ) bk∗ bk ,

.

dom(H0,Λ,μ ) ⊂ FΛ . (2.33)

k∈Λ∗

Here one-particle kinetic-energy operator spectrum .{εk = k 2 }k∈Λ∗ , where the dual to .Λ set is : Λ∗ = {kj = 2π nj /V αj : nj ∈ Z}d=3 j =1 and εk =

d

.

kj2 .

(2.34)

j =1

We denote by .bk := b(φkΛ ) and .bk∗ := (b(φkΛ ))∗ the boson annihilation and creation operators in mode .k ∈ Λ∗ in the√ Fock space .FΛ . They are indexed by the ortho-normal basis .{φkΛ (x) = eikx / V }k∈Λ∗ ⊂ L2 (Λ) generated by the eigenfunctions of the self-adjoint one-particle kinetic-energy operator .(−Δ)p.b.c. in 2 .L (Λ). Formally annihilation and creation operators satisfy the Canonical Commutation Relations (CCR): .[bk , bk∗' ] = δk,k ' 1. Then .Nk = bk∗ bk is occupation-number  operator of the one-particle state .φkΛ and .NΛ = k∈Λ∗ Nk is the total-number operator in .FΛ . If we define operator of the Bose-field:

x I→ b(x) :=

.

bk φkΛ (x) ,

x ∈ Λ.

(2.35)

k∈Λ∗

then the boson annihilation operator  bk =

.

1 dx b(x)) φkΛ (x) = √ V Λ



dx b(x)) e−i kx ,

k ∈ Λ∗ .

(2.36)

Λ

0 If we denote by .ωβ,μ,Λ (·) the grand-canonical Gibbs state (2.1) of PBG, which is generated by density matrix (2.2) for (2.33), then the problem of existence of conventional Bose-Einstein condensation is related to solution of the equation

ρ=

.

1 1

1 0 , ωβ,μ,Λ (Nk ) = β(εk −μ) − 1 V V e ∗ ∗ k∈Λ

(2.37)

k∈Λ

for a given total particle density .ρ in .Λ. Note that by (2.34) the thermodynamic limit Λ ↑ R3 in the right-hand side of (2.37):

.

On Ergodic States

441

I (β, μ) = lim

.

Λ

 1 0 1 1 , d 3 k β(ε −μ) ωβ,μ,Λ (Nk ) = 3 k 3 V (2π ) e −1 R ∗

(2.38)

k∈Λ

exists for any .μ < 0. It reaches its (finite) maximal value .I (β, μ = 0) = ρc (β), which is called the critical particle density for a given temperature. The existence of finite .ρc (β) triggers (by means of the saturation mechanism) a non-zero BEC .ρ0 (β, ρ) := ρ − ρc (β), when the total particle density .ρ > ρc (β). To this end we note that for .α1 < 1/2, the whole condensate is sitting in the one-particle ground-state mode .k = 0:

−1 1 −β μΛ (β,ρ≥ρc (β)) 1 0 e ωβ,μ,Λ (N0 ) = lim −1 Λ V Λ V 1 1 + o(1/V ) , μΛ (β, ρ ≥ ρc (β)) = − V β (ρ − ρc (β)) ρ0 (β, ρ) = ρ − ρc (β) = lim

.

where .μΛ (β, ρ) is a unique solution of Eq. (2.37). This is a well-known conventional (or the type I [8]) condensation. In particular, in this case it make sense the ODLRO for the Bose-field (2.35). Indeed, by Definition 3 one gets for the product of spacial averages (2.17) of Bose-fields .b∗ (x) and .b(x):  0 . lim ωβ,μ,Λ Λ

1 V



1 dx b (x) V Λ ∗



 Λ

0 dx b(x) = lim ωβ,μ,Λ ( Λ

b0∗ b0 ) = ρ0 (β) , V (2.39)

that is, the ODLRO (2.32) is nontrivial and coincides with the condensate density. For .α1 = 1/2 (the Casimir box, see [8] and [26, 28]) one observes the case of infinitely many macroscopically occupied levels, which is called the type II generalised Bose-condensation (gBEC). On the other hand, when .α1 > 1/2 (the van den Berg-Lewis-Pulé box [8], see [44], Section 3.1) one infers that .

0 lim ωβ,μ,Λ ( Λ

−1 bk∗ bk 1 β(εk −μΛ (β,ρ)) e ) = lim −1 = 0 , ∀ k ∈ Λ∗ , Λ V V

(2.40)

i.e., there is no macroscopic occupation of any mode for any value of particle density ρ. But a gBEC of type III does exist in the following sense:

.

ρ − ρc (β) = lim lim

.

ϵ→+0 Λ

1 V





−1 eβ(εk −μΛ (β,ρ)) − 1 , for ρ > ρc (β) .

{k∈Λ∗ ,‖k‖≤ϵ}

(2.41) Note that (2.39) and (2.40) imply the ODLRO, whereas the type III condensation (2.41) not.

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We comment that this unusual condensation is not exclusively due to the special geometry .α1 > 1/2, see also [3]. In fact the same phenomenon of the gBEC (type III) happens due to interaction in the model with Hamiltonian: HΛ =



.

εk bk∗ bk +

k∈Λ∗

a ∗ ∗ bk bk bk bk , a > 0 , 2V ∗

(2.42)

k∈Λ

see [44], Section 3.1. Another case of unusual Bose-condensation due to interaction is known as non-conventional (or dynamical) BEC, see [44], Chapter 5, and review [43]. This condensation exists because of interaction, i.e., it disappears, when interaction is turned off. The concepts of ODLRO and SSB are applicable in this case and the Bogoliubov quasi-average method is an important part of technical tools for analysis and proofs. For example it allowed to solve the Bogoliubov model of Weakly Imperfect Bose-gas [44]. Examples of different types of condensations show that connections between BEC, ODLRO, and SSB are a subtle matter. They motivate and bolster a relevance of the Bogoliubov quasi-average method [4–6], that we shall discuss in the next two sections.

3 Selection of Pure States by the Bogoliubov Quasi-Averages: Spin Systems Considering further the simple example of spin system (2.25) for the sake of argument, at least two methods of selecting pure states may be suggested: (1) By taking in (2.1), (2.2) .HΛ with special boundary conditions (b.c.), i.e., upon imposing on the boundary .∂Λ of .Λ |n)x such that σx |n)x = |n)x .

.

(3.1)

The above choice leads, presumably, to the limiting states .ωβ,n in (2.30). (2) By replacing in (2.1), (2.2) .HΛ by the quasi-Hamiltonian HΛ,B := HΛ + HΛB ,

.

(3.2)

with the symmetry-breaking vector field .B n directed along the unit vector n: HΛB = −B n ·



.

σx , B > 0 .

(3.3)

x∈Λ

We take .B → +0 after the thermodynamic limit .V → ∞. This method, known as the Bogoliubov quasi-averages, is currently employed as a trick, i.e., without

On Ergodic States

443

 explicit connection to ergodic states. The quantity . x∈Λ σx (the magnetization) in the symmetry-breaking field is known as the order parameter. As spelled out in (3.3), it is appropriate to the Heisenberg ferromagnet (2.25) and for the XY model, but not for the antiferromagnet, in which case the order parameter should be replaced by the sub-lattice magnetization . x∈Λ∩A σx , where .Zd = A ∪ B, .A, B denoting two disjoint sublattices. If we consider first .0 < β < ∞, .G = SO(3) and .HΛ the Hamiltonian (2.25) (or its antiferromagnetic or XY analog), with free or periodic b.c., then .HΛ is Ginvariant, and thus .ωβ,Λ , defined by (2.1),(2.2), is also G-invariant. Taking, now, .HΛ with the b.c. (2), both .HΛ and .ωβ,Λ are not G-invariant. Consider, now, .β = ∞, i.e., the ground state, with .HΛ given by (2.25), defined with free or periodic b.c.. Again, .HΛ is invariant under G, and we may regard a ground state ω∞,Λ (·) = (ΩΛ , · ΩΛ ) ,

(3.4)

|ΩΛ = ⊗x∈Λ |n)x .

(3.5)

.

with .

Then, clearly, .ω∞,Λ as well as its infinite volume counterpart is not G-invariant. Note that (3.4) leads, however, presumably to the ergodic states .ω∞,n in the decomposition (2.30), when taking the weak*-limit as .Λ ↗ Z3 . If we take, however, the weak*-limit, as .β → ∞ along a subsequence, of .ωβ , it may be conjectured that the G-invariant ground state  ω∞ :=

.

dμn ω∞,n ,

is obtained. The limits .V → ∞ and .β → ∞ are not expected to commute, and we believe, in consonance with the third principle of thermodynamics [40], that it is more adequate, both physically and mathematically, to regard the states .ωβ for .0 < β < ∞ as fundamental, with ground states defined as their (weak*) limit as .β → ∞ (along a subsequence or subnet). In this sense, the assertion found in most textbooks, see also [24] beginning of Section 2, that SSB occurs when the Hamiltonian is invariant, but not the state, is not correct, or, at least, not precise. Note, however, that, in the textbooks, “state” is understood as the ground state or the vacuum state, but not as the thermal state, for which the equivalence between the invariance of the Hamiltonian and the state is essentially obvious. If one uses the method (2) of Bogoliubov quasi-averages, such difficulties do not appear, because .ωβ,n is thereby directly connected to .ω∞,n for each .n. Moreover, as we motivated at the end of Sect. 2 by the example of gBEC, the quasi-average method is even indispensable for quantum continuous Bose-systems. An example of its use appears in the next Sect. 4. See also the conclusion, Sect. 5.

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We note that for quantum continuous many-body systems or relativistic quantum field theory imposition of boundary conditions is very questionable, or even not feasible. The proof of the method (2) for the ferromagnet follows [24], but using Bloch coherent states, instead of Glauber coherent states, in the manner of Lieb’s classic work on the classical limit of quantum spin systems [21]. It will not be spelled out here, because the next section will be devoted to a similar proof in the case .G = U (1) and Boson systems, but we note the result: Proposition 2 The ergodic states .ωβ,n in the decomposition (2.30) may be obtained by the Bogoliubov quasi-average method: ωβ,n = lim

.

lim ωβ,Λ,n

(3.6)

TrHΛ (exp(−βHΛ,B )A) , TrHΛ exp(−βHΛ,B )

(3.7)

B→+0 V →∞

where ωβ,Λ,n (A) ≡

.

with .A ∈ B(HΛ ), and .HΛ,B is defined by (3.2), (3.3) for the ferromagnet (2.25). The limit (3.6) is taken along a (double) subsequence of the variables .(B, V ). For A of the form (2.27), the actual double limit in (3.6) exists, and, if ODLRO holds in the form (2.32), then SSB in the sense of Definition 2, cf. (2.14) and (2.30), holds for the states (3.6). The identification of .ωβ,n in (3.6) with those occurring in the decomposition (2.29) is possible by the unicity of the ergodic decomposition, in view of the fact that the spin algebra is asymptotically abelian for the space translations, see [10], pp 380–381. Since for KMS states the ergodic decomposition coincides with the central decomposition, the extension of the states to elements of the center is also unique.

4 Continuous Boson Systems: Quasi-Averages, Condensates, and Pure States We now study the states of Boson systems, and, for that matter, assume, together with Verbeure ([37], Ch.4.3.2) that they are analytic in the sense of [11], Ch.5.2.3. We start, with the self-adjoint Hamiltonian for Bosons in a cubic box .Λ of side L and volume .V = L3 in the boson Fock space .FΛ := F (L2 (Λ)): HΛ,μ = H0,Λ,μ + VΛ ,

.

where

(4.1)

On Ergodic States

445

VΛ =

.

1

∗ ∗ v(p)bk+p bq−p bk bq , V

(4.2)

k,p,q

with periodic b.c., .h¯ = 2m = 1, and .k, p, q ∈ Λ∗ . Here .Λ∗ (2.34) is dual (with respect to Fourier transformation) set corresponding to .Λ. Here .ν is the Fourier transform of the two-body potential .v(x), with bound |v(k)| ≤ v(0) < ∞ ,

(4.3)

.

and H0,Λ,μ =



.

k 2 bk∗ bk − μNΛ ,

(4.4)

k

NΛ =



.

bk∗ bk ,

(4.5)

k

with .[bk , bl∗ ] = δk,l the second quantised annihilation and creation operators, cf. (2.35) and (2.36). The quasi-Hamiltonian corresponding to (3.2) is taken to be λ

HΛ,μ,λ = HΛ,μ + HΛφ ,

(4.6)

.

with the symmetry-breaking field analogous to (3.3) given by Bogoliubov [5] λ

HΛφ := −

√

.

V (λ¯ φ b0 + λφ b0∗ ) ,

λ

dom(HΛφ ) ⊂ F0 ,

(4.7)

where .F0 ⊂ FΛ is the zero-mode boson subspace, and λφ := λ exp(iφ) with λ ≥ 0 , and arg(λφ ) = φ ∈ [0, 2π ) .

.

(4.8)

Note that linear term (4.7) breaks the gauge invariance (with respect to: .bk → bk e−i s , s ∈ R) of the original Hamiltonian (4.4) in the zero mode .k = 0. We remind that the group of gauge transformations .{τs }s∈[0,2π ) can be defined as transformations τs (b∗ (f )) = exp(i s) b∗ (f ) ,

.

τs (b(f )) = exp(−i s) b(f ) ,

(4.9)

where .b∗ (f ), b(f ) are creation and annihilation operators smeared over testfunctions f , cf. (2.36). Then .U : s I→ exp(i s b∗ (f ) b(f )) is the unitary implementation of this group on the boson Fock space .FΛ : .U (s) b(f ) U (s)∗ = exp(−i s) b(f ). We take initially .λ > 0 and consider first the perfect Bose-gas to define λ

H0,Λ,μ,λφ = H0,Λ,μ + HΛφ .

.

(4.10)

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W. F. Wreszinski and V. A. Zagrebnov

Then we may present (4.10) as H0,Λ,μ,λφ := H0,λφ + Hk/=0 ,

.

where H0,λφ := −μ b0∗ b0 −

.

√ V (λ¯ φ b0 + λφ b0∗ )

(4.11)

We introduce the canonical (that is, preserving CCR) shift transformation: √ λφ V  , .b0 := b0 + μ

(4.12)

without altering the non-zero modes in .Hk/=0 , and assume henceforth .μ < 0. b0 e−i s (4.38), of This transformation restores the gauge invariance for . b0 →  Hamiltonian .H0,Λ,μ,λφ since (4.11) takes the form λ2 0,λφ := −μ  . b0∗  H b0 + V μ

.

(4.13)

Note that canonical transformation (4.12) has unitary implementation .U : α I→ exp(i α [(b0∗ − b0 )/ i]), on the Fock space .FΛ , that is, .U (α) b0 U (α)∗ = b0 + α. Hence, the grand partition function .ΞΛ (i.e. the trace (2.3)) is invariant with respect of this transformation. Then .ΞΛ splits into product of traces over the zero-mode and the remaining modes. As a consequence, due to (4.13) we obtain ΞΛ (β, μ, λ) = (1 − exp(βμ))−1 exp(−

.

βλ2 V ) ΞΛ' , μ

(4.14)

ϵk = k 2 .

(4.15)

where 

ΞΛ' :=

.

(1 − exp(−β(ϵk − μ)))−1 ,

k/=0

The corresponding thermodynamic potential is the pressure: pβ,μ,Λ,λ =

.

1 ln ΞΛ (β, μ, λ) . βV

(4.16)

Recall that the finite-volume grand-canonical Gibbs state for the perfect Bose-gas with Hamiltonian (4.10) is 0 ωβ,μ,Λ,λ (·) := φ

.

1 −βH0,Λ,μ,λφ TrFΛ [e (·)] , ΞΛ

(4.17)

On Ergodic States

447

see Sect. 2. One can use canonical transformation (4.12) for a straightforward calculation of expectation of operators .b0 and .b0∗ . Indeed, by the canonical transformation we obtain equation √ 1 0,Λ,μ,λ −β H 0 φ  ωβ,μ,Λ,λ (b0 + λφ V /μ) = TrFΛ [e b0 ] = 0 , φ ΞΛ

.

(4.18)

0,λφ + Hk/=0 , and the last equality is because of 0,Λ,μ,λφ := H where Hamiltonian .H its gauge invariance. As a consequence √ λ¯ φ 0 (b0∗ ) = − V and ωβ,μ,Λ,λ . φ μ

√ λφ 0 ωβ,μ,Λ,λ (b0 ) = − V φ μ

.

(4.19)

One can consider (4.16) as generating function and differentiate it with respect to .μ < 0. Then from (4.14)–(4.17) it follows that the mean particle density 1 + V (exp(−βμ) − 1)

0 ρβ,μ,Λ,λφ := ωβ,μ,Λ,λ (NΛ /V ) = φ

|λφ |2 1 1

. + V exp(β(ϵk − μ)) − 1 μ2

.

+

k/=0

(4.20) Equation (4.20) is the starting point for our analysis of SSB and Bose-condensation. To this end let  dk .ρc (β) := (exp(βϵk ) − 1)−1 . (4.21) 2π 3 Proposition 3 Let .0 < β < ∞ be fixed. Then, for each value of particle density ρ = ρβ,μ,Λ,λφ such that

.

ρc < ρ < ∞ ,

(4.22)

.

and for each .|λφ | = λ > 0 , .V < ∞ , there exists a unique solution of (4.20), which has the form μ(β, λ, ρ, V ) = − √

.

λ + α(λ, V ) , ρ − ρc (β)

(4.23)

where α(λ, V ) ≥ 0 , and

.

lim

lim

λ→+0 V →∞

α(λ, V ) = 0. λ

(4.24)

448

W. F. Wreszinski and V. A. Zagrebnov

Remark 4 We skip the proof of this lemma, but we note that besides the cube .Λ, it is also true for the case of three-dimensional anisotropic parallelepiped .Λ := V α1 × V α2 × V α3 , with periodic boundary condition (p.b.c.) and .α1 ≥ α2 ≥ α3 , .α1 + α2 + α3 = 1, i.e. the volume .|Λ| = V , see [18], Sec.4, and [41], Sec.2.2. As a consequence of Proposition 3 by (4.19) and (4.23) we infer lim

.

√ √ 0 lim ωβ,μ,Λ,λ (b0 / V ) = ρ0 exp(i φ) , φ

λ→+0 V →∞

(4.25)

where .ρ0 (β, ρ) := ρ − ρc (β) is the Bose-condensate density of the perfect Bosegas. We see therefore that the phase .φ in (4.19) remains in (4.25) even after the limit .λ → +0. Consequently, the quasi-average double limit defines a family of the limiting Gibbs states for the perfect Bose-gas: 0 ωβ,μ,φ := lim

.

0 lim ωβ,μ,Λ,λ , φ

λ→+0 V →∞

φ ∈ [0, 2π ) ,

(4.26)

where the limit along a subnet exists by the weak*-compactness, [10, 17]. 0 The limits (4.25), (4.26) show that for .ρ0 (β, ρ) > 0 the Gibbs states .ωβ,μ,φ are not gauge invariant. Assuming that they are ergodic states in the ergodic 0 , which we shall prove next for the interacting system, it decomposition of .ωβ,μ follows that for the perfect Bose gas the BEC is equivalent to the SSB. But It is, however, illuminating to see this also in the case of another explicit mechanism of appearance the phase .φ. This mechanism is related to solution (4.23), Proposition 3, and yields that the chemical potential remains proportional to .|λφ | even after the thermodynamic limit. As we shall see below, that this property persists for the interacting system (4.1)–(4.5). Remark 5 Note that results of Proposition 3 and Remark 4 are independent of the anisotropy, i.e. of whether the condensation for .λ = 0 is in single mode (.k = 0), or it is extended as the gBEC-type III, Sect. 2. This means that the Bogoliubov quasiaverage method allows to study the question about equivalence between .(BEC)qa , .(SSB)qa and .(ODLRO)qa if they are defined via one-mode quasi-average for .‖k‖ ≥ 0. To check this assertion we re-exemine the prefect Bose-gas (2.33) with symmetry breaking sources (4.7) in a single mode .q ∈ Λ∗ : H0,Λ,μ,ν := H0,Λ,μ −

.

√   V ν bq + ν bq∗ ,

μ < 0, ν ∈ C.

(4.27)

Then for a fixed density .ρ, the grand-canonical condensate Eq. (2.37) for (4.27) takes the following form (cf. calculation of (4.20):

On Ergodic States

449

ρ = ρΛ (β, μ, ν) :=

1 0 ωβ,μ,Λ,ν (bk∗ bk ) = V ∗ k∈Λ

.

1 β(εq −μ) |ν| 2 1 (e − 1)−1 + + V V (εq − μ) 2

k∈Λ∗ \q

1 eβ(εk −μ) − 1

(4.28) .

According to the quasi-average method, to investigate a possible condensation, one must first take the thermodynamic limit in the right-hand side of (4.28), and then switch off the symmetry breaking source: .ν → 0. Recall that the critical density, which defines the threshold of boson saturation is equal to .ρc (β) = I (β, μ = 0) (2.38), where .I (β, μ) = limΛ ρΛ (β, μ, ν = 0). Given that .μ < 0, we have to distinguish two cases: (1) Let .q ∈ Λ∗ be such that .limΛ εq > 0. Then we obtain from (4.28) the limiting condensate equation ρ = lim lim ρΛ (β, μ, ν) = I (β, μ) ,

.

ν→0 Λ

i.e., the quasi-average coincides with the average. Hence, we return to the analysis of the condensate Eq. (4.28) for .ν = 0. This leads to finite-volume solutions .μΛ (β, ρ) and consequently to all possible types of condensation as a function of anisotropy .α1 , see Sect. 2 for details. (2) On the other hand, if .q ∈ Λ∗ is such that .limΛ εq = 0, then thermodynamic limit in the right-hand side of the condensate Eq. (4.28) yields: ρ = lim ρΛ (β, μ, ν) = I (β, μ) +

.

Λ

|ν| 2 . μ2

(4.29)

Now, if .ρ ≤ ρc (β), then the limit of solution of (4.29): .limν→0 μ(β, ρ, ν) = μ0 (β, ρ) < 0, where .μ(β, ρ, ν) = limΛ μΛ (β, ρ, ν) < 0 is thermodynamic limit of the finite-volume solution of condensate Eq. (4.28). As a result, there is no condensation in any mode. But if .ρ > ρc (β), then .limν→0 μ(β, ρ, ν) = 0 and the density of condensate is |ν| 2 . ν→0 μ(β, ρ, ν) 2

ρ0 (β, ρ) = ρ − ρc (β) = lim

.

(4.30)

Note that expectation of the particle density in the q-mode (see (4.28)) is 0 ωβ,μ,Λ,ν (bq∗ bq /V ) =

.

1 β(εq −μ) |ν| 2 (e . − 1)−1 + V (εq − μ) 2

Then by (4.30) the corresponding Bogoliubov quasi-average for .bq∗ bq /V is equal to

450

W. F. Wreszinski and V. A. Zagrebnov 0 ρ − ρc (β) = lim lim ωβ,μ (bq∗ bq /V ) = Λ (β,ρ,ν),Λ,ν ν→0 Λ

.

lim lim

ν→0 Λ

1 β(εq −μΛ (β,ρ,ν)) |ν| 2 (e − 1)−1 + , V (εq − μΛ (β, ρ, ν)) 2

(4.31)

where .μΛ (β, ρ, ν) < 0 is a unique solution of the condensate Eq. (4.28) for .ρ > ρc (β). On account of (4.28) and (4.30) one gets .μ(β, ρ, ν /= 0) < 0. Hence, for any .k /= q such that .limΛ εk = 0 we get .

0 lim lim ωβ,μ (bk∗ bk /V ) = lim lim Λ (β,ρ,ν),Λ,ν

ν→0 Λ

ν→0 Λ

1 1 =0. β(ε −μ (β,ρ,ν))) Λ k V e −1 (4.32)

Consequently, for any .α1 the quasi-average condensation .(BEC)qa occurs only in one mode (type I), whereas for .α1 > 1/2 the BEC is of the type III, see Sect. 2. Similarly, diagonalisation by the canonical shift transformation, see (4.12) and (4.30), allows to apply the quasi-average method to calculate a nonvanishing for .ρ > ρc (β) gauge-symmetry breaking .(SSB)qa : .

 ν = ei arg(ν) ρ − ρc (β) , ν→0 μ(β, ρ, ν) (4.33)

√ 0 lim lim ωβ,μ (bq / V ) = lim Λ (β,ρ,ν),Λ,ν

ν→0 Λ

along the path .{ν = |ν|ei arg(ν) ∧ |ν| → 0}. Then by inspection of (4.31) and (4.33) we find that .(BEC)qa and .(SSB)qa are equivalent: √ √ 0 ∗ 0 lim lim ωβ,μ (b / V ) ω (b / V) = q q β,μ (β,ρ,ν),Λ,ν (β,ρ,ν),Λ,ν Λ Λ

ν→0 Λ .

0 = lim lim ωβ,μ (bq∗ bq /V ) = ρ − ρc (β) . Λ (β,ρ,ν),Λ,ν

(4.34)

ν→0 Λ

Note that by (2.39) the .(SSB)qa and .(BEC)qa are in turn equivalent to (ODLRO)qa , whereas for .α1 > 1/2 and any .ρ, .q ∈ Λ∗ one gets:

.

.

0 lim ωβ,μ (bq∗ bq /V ) = Λ (β,ρ,ν=0),Λ,ν=0 Λ

√ √ 0 0 lim ωβ,μ (bq / V ) = 0 , (bq∗ / V ) ωβ,μ Λ (β,ρ,0),Λ,0 Λ (β,ρ,0),Λ,0 Λ

because the conventional condensation is of the type III (2.40). We now consider the interacting case (4.1)–(4.5). We say (cf Sect. 2) that the interacting Bose-gas undergoes the zero-mode Bose-Einstein condensation (BEC) (and/or ODLRO, cf. (2.39)) if .

lim ωβ,μ,Λ (b0∗ b0 /V ) = ρ0 > 0 .

V →∞

(4.35)

On Ergodic States

451

For this and the forthcoming definitions, we are referring to the full interacting Bose gas (4.1)–(4.3), with .ω, which replaces .ω0 . The corresponding definitions for the general case of the quantities .ωβ,μ,Λ,λφ and .ωβ,μ,φ are the obvious analogues of (4.17) and (4.26), with .H0,Λ,μ,λφ (4.27) replaced by .HΛ,μ,λφ (4.6). Remark 6 The famous Bogoliubov approximation of replacing spacial averages ηΛ (b) and .ηΛ (b∗ ) (2.17) by c-numbers [42, 44] will be a key instrument. It was proved by Ginibre [14], Lieb et al. [23, 24] and Sütö [35], but we shall rely on the method of [24], which uses the Berezin-Lieb inequality [21]. Although in [24] the phase .φ = 0, their results can be easily extended to general case (4.8) using simple substitutions: .b0 → b0 exp(−i φ), .b0∗ → b0∗ exp(i φ), motivated by the symmetrybreaking term (4.7).

.

Recall that the boson Fock space has tensor structure: .FΛ ≃ F0 ⊗ F ' , where .F0 denotes the zero-mode subspace (4.7) and 

F ' :=

Fk .

.

(4.36)

k∈Λ∗ \{0}

Let .z ∈ C be a complex number, and .|z〉 := exp(−|z|2 /2 + z b0∗ ) |0〉 be normalised coherent state vector in subspace .F0 ⊂ FΛ , which is generated from vacuum: .|0〉 ∈ FΛ , by the zero-mode creation operators. Note that .b0 |z〉 = z |z〉, ∗ 2 .〈z |b b0 | z〉 = |z| . Then owing to partial trace and partial inner product for the 0 pressure .pβ,μ,λ,Λ of the system with Hamiltonian (4.6): exp(βVpβ,μ,λ,Λ ) = ΞΛ (β, μ, λ) = TrF exp(−β(HΛ,μ,λ ) =  . = d 2 z TrF ' 〈z | exp(−βHΛ,μ,λ ) | z〉 ,

(4.37)

C

 where .d 2 z = dRe(z) dIm(z) and . C d 2 z | z〉〈z | = 1. On account of coherent vector .|z〉 by the partial inner product one defines the lower symbol of operator .HΛ,μ,λ : '

(HΛ,μ,λ ) (z) := 〈z | HΛ,μ,λ | z〉 ,

.

'

dom((HΛ,μ,λ ) (z)) ⊂ F ' .

(4.38)

'

Then we introduce a lower pressure .pβ,Λ,μ,λ : '

.

'

exp(βVpβ,Λ,μ,λ ) := ΞΛ (β, μ, λ) :=



'

C

d 2 z TrF ' exp(−β(HΛ,μ,λ ) (z)) , (4.39)

and consider the probability density Wμ,Λ,λ (z) := ΞΛ (β, μ, λ)−1 TrF ' 〈z | exp(−βHΛ,μ,λ ) | z〉 .

.

(4.40)

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W. F. Wreszinski and V. A. Zagrebnov

√ As it is proved in [24], the density of distributions .{Wμ,Λ,λ (ζ V )}V converges for almost all .λ > 0, as .V → ∞, to .δ-density at the point √ ζmax (λ) = lim zmax (λ)/ V .

.

V →∞

'

Here .zmax (λ) is maximiser of the partition function .TrH ' exp(−β(HΛ,μ,λ ) (z)), cf. (4.39). Although in [24] the phase .φ = 0, their results to the general case (4.8) can be obtained using the simple substitutions: .b0 → b0 exp(iφ), ∗ ∗ .b 0 → b0 exp(−iφ) motivated by (4.7). As a consequence, expression (34) in [24] can be re-written as .

lim ωβ,μ,Λ,λ (ηΛ (b0∗ exp(iφ)) = lim ωβ,μ,Λ,λ (ηΛ (b0 exp(−iφ))

V →∞

V →∞

∂p(μ, λ) , = ζmax (λ) = ∂λ

(4.41)

and consequently .

lim ωβ,μ,Λ,λ (ηΛ (b0∗ )ηΛ (b0 )) = |ζmax (λ)|2 .

V →∞

(4.42)

Here above, p(β, μ, λ) = lim pβ,μ,Λ,λ ,

.

V →∞

(4.43)

is the pressure in the thermodynamic limit. Equality (4.41) follows from the convexity of .pβ,μ,Λ,λ in .λ by the Griffiths lemma [15]. As it is shown in [24] the pressure .p(β, μ, λ) is equal to '

'

p(β, μ, λ) = lim pβ,μ,Λ,λ .

.

V →∞

(4.44)

''

As well as it is also equal to the pressure .p(β, μ, λ) , which is the thermodynamic limit of the pressure associated to the upper symbol of .HΛ,μ,λ . It is crucial in the proof of [24] that all of these three pressures coincide with ' .pmax (β, μ, λ), which is the pressure associated to .maxz Tr H ' exp(−β(HΛ,μ,λ ) (z)). Theorem 1 Consider the system of interacting Bosons (4.1)–(4.8). If the system displays ODLRO in the sense of (4.37), the limit .ωβ,μ,φ := limλ→+0 limV →∞ ωβ,μ,Λ,λφ , on the set .{η(b0∗ )m η(b0 )n }m,n=0,1 exists and satisfies ωβ,μ,φ (η(b0∗ )) =

.

ωβ,μ,φ (η(b0 )) =

.

√

ρ0 exp(iφ) ,

√ ρ0 exp(−iφ) ,

(4.45) (4.46)

On Ergodic States

453

together with ωβ,μ,φ (η(b0∗ )η(b0 ) = ωβ,μ ((η(b0∗ )η(b0 )) = ρ0 ∀φ ∈ [0, 2π ) ,

.

(4.47)

and ωβ,μ =

.

1 2π



2π

dφ ωβ,μ,φ .

(4.48)

0

On the Weyl algebra the limit defining .ωβ,μ,φ , φ ∈ [0, 2π ) exists along a net in the .(λ, V ) variables, and defines ergodic states coinciding with those states that explicitly constructed in Theorem 2. Conversely, if SSB occurs in the special sense that (4.41) and (4.42) hold, with .ρ0 /= 0, then ODLRO in the sense of (4.25) takes place. Proof We need only prove the direct statement, because the converse follows by applying the Schwarz inequality to the states .ωβ,μ,φ , together with the forthcoming (4.55). We thus prove ODLRO .⇒ SSB. We first assume that some state .ωβ,μ,φ0 , φ0 ∈ [0, 2π ) satisfies ODLRO. Then by (4.42), .

lim

lim ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) = lim |ζmax (λ)|2 =: ρ0 > 0 .

λ→+0 V →∞

λ→+0

(4.49)

The above limit exists by the convexity of .p(μ, λ) in .λ and (4.25). By virtue of (4.49), .

∂p(μ, λ) /= 0 . λ→+0 ∂λ lim

(4.50)

At the same time, (4.41) shows that all states .ωβ,μ,φ satisfy (4.49). Thus, SSB is broken in the states .ωβ,μ,φ , φ ∈ [0, 2π ). We now prove that the original assumption (4.37) implies that all states .ωβ,μ,φ , φ ∈ [0, 2π ) exhibit ODLRO. Gauge invariance of .ωβ,μ,Λ (or equivalently .HΛ,μ ) yields, by (4.7), (4.38), ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) = ωβ,μ,Λ,−λ (η(b0∗ )η(b0 )) .

.

(4.51)

Again by (4.7), (4.23) and gauge invariance of .HΛ,μ , .

lim

λ→−0

∂p(μ, λ) ∂p(μ, λ) = − lim , λ→+0 ∂λ ∂λ

and, since by convexity the derivative .∂p(μ, λ)/∂λ is monotone increasing, we find .

lim

λ→+0

∂p(μ, λ) √ = lim ζmax (λ) = ρ0 , λ→+0 ∂λ

(4.52)

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W. F. Wreszinski and V. A. Zagrebnov

.

lim

λ→−0

∂p(μ, λ) √ = − lim ζmax (λ) = − ρ0 . λ→+0 ∂λ

(4.53)

Again by (4.51), lim

.

lim ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) = lim

λ→−0 V →∞

lim ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) .

λ→+0 V →∞

(4.54) By Lieb et al. [24], the weight .Wμ,λ is, for .λ = 0, supported on a disc with radius equal to the right-derivative (4.50). Convexity of the pressure as a function of .λ implies

.

∂p(μ, λ− 0) ∂λ− 0

∂p(μ, λ+ ∂p(μ, λ) ∂p(μ, λ) 0) ≤ lim ≤ , + λ→−0 λ→+0 ∂λ ∂λ ∂λ0

≤ lim

+ for any .λ− 0 < 0 < λ0 . Therefore, by the Griffiths lemma (see e.g. [15, 24]) one gets

.

lim

lim ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) ≤ lim ωβ,μ,Λ (

λ→−0 V →∞

V →∞

≤ lim

b0∗ b0 ) V

lim ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) .

λ→+0 V →∞

(4.55) Then (4.54) and (4.55) yield .

lim ωβ,μ,Λ (

V →∞

b0∗ b0 ) = lim lim ωβ,μ,Λ,λ (η(b0∗ )η(b0 )) ∀φ ∈ [0, 2π ) . λ→+0 V →∞ V (4.56)

This proves that all .ωβ,μ,φ , φ ∈ [0, 2π ) satisfy ODLRO, as asserted. By (4.41) and (4.52) one gets (4.45) and (4.46). Then (4.48) is a consequence of the gauge-invariance of .ωβ,μ . Ergodicity of the states .ωβ,μ,φ , φ ∈ [0, 2π follows from (4.56) and (4.45), (4.46). An equivalent construction is possible using the Weyl algebra instead of the polynomial algebra, see [37], pg. 56 and references given there for Theorem 2 and similarly we could have proceeded so here. The limit along a subnet in the .(λ, V ) variables exists by weak* compactness, and, by asymptotically abelianness of the Weyl algebra for space translations (see, e.g., [11], Example 5.2.19), the ergodic decomposition (4.48), which is also a central decomposition, is unique. Thus, the .ωβ,μ,φ , φ ∈ [0, 2π ) coincide with the states constructed in Theorem 2. ⨆ ⨅ Remark 7 Our Remark 4.3 and Theorem 1 elucidate a problem discussed in [24]. In this paper the authors defined a generalised Gauge Symmetry Breaking via quasi-average .(GSB)qa , i.e. by .limλ→+0 limV →∞ ωβ,μ,Λ,λ (ηΛ (b0 )) /= 0. (If

On Ergodic States

455

it involves other than gauge group, we denote this by .(SSB)qa .) Similarly they modified definition of the one-mode condensation denoted by .(BEC)qa (4.49), and established the equivalence: .(GSB)qa ⇔ (BEC)qa . They asked whether .(BEC)qa ⇔ BEC ? We show that .(GSB)qa coincides with GSB (.λ = 0), and that BEC is indeed equivalent to .(BEC)qa . Remark 8 The states .ωβ,μ,φ in Theorem 1 have the property (2) of Theorem 2, i.e., if .φ1 /= φ2 , then .ωβ,μ,φ1 /= ωβ,μ,φ2 . By a theorem of Kadison [19], two factor states are either disjoint or quasi-equivalent (see Remark 3.1 and references given in [19]), and thus the states .ωβ,μ,φ for different .φ are mutually disjoint. This fact has a simple explanation: only for a finite system is the Bogoliubov transformation (4.12) (which also applies to the interacting system), which connects different .φ, unitary: for√infinite systems one has to make an infinite change of an extensive observable .b0 V , and mutually disjoint sectors result. This phenomenon also occurs with regard to the magnetization in quantum spin systems, in correspondence to (2.30) and it is in this sense that the word “degeneracy” must be understood (compare with the discussion in [4]).

5 Concluding Remarks In this paper, we reexamined the issue of ODLRO versus SSB by the method of Bogoliubov quasi-averages, commonly regarded as a symmetry-breaking trick. We showed that it represents a general method of construction of extremal, pure or ergodic states, both for quantum spin systems (Proposition 2) and many-body Boson systems (Theorem 1). The breaking of gauge symmetry in the latter has some analogy with the breaking of gauge and .γ5 invariance in the Schwinger model (quantum electrodynamics of massless electrons in two dimensions) [25], in which the vacuum state decomposes in a manner similar to (4.48). We believe, and argued so in Sect. 3, that the quasi-average method is the only universally applicable method, in particular to relativistic quantum field theory, to which the imposition of classical boundary conditions is bound to be inconsistent with the general principles of local quantum theory, as in the case of the Casimir effect [20]. A general necessary feature for the applicability of the Bogoliubov method is the existence of an order parameter. In the two examples treated, the Heisenberg ferromagnet (Sect. 2, see also Remark 2 concerning order parameters for quantum spin systems in the general case) and many-body Boson systems (Sect. 4), the respective symmetry-breaking fields (3.3) and (4.7) are qualitatively different. Note that (3.3) commutes with .HΛ and the corresponding order parameter, the magnetization, is physically measurable. Whereas (4.7) does not commute with .HΛ − μNΛ (even in the free-gas case!), and the order parameter involves a phase by (4.45), (4.46), which, at first glance, is not physically measurable. It has been observed, however, in the interference of two condensates of different phases [1] , [3], in the case of trapped gases. In the latter case, Condensation takes place at .k /= 0,

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and the version of Theorem 1 due to Pulè et al [29] is the relevant one. Finally, in the quantum spin case there is a residual symmetry (Remark 1, but none, of course, in the Boson case. These remarks exemplify the rather wide diversity of types of the Bogoliubov quasi-avearge, which make its conjectured universal applicability further plausible, see e.g. random boson systems [18]. As remarked by Swieca [36], it is the fluctuations occurring all over space which do not allow to take the “charge” (e.g. (2.26)) in the limit .V → ∞ as a well-defined operator (this would, in particular, contradict (2.23)), even if a meaning has been given to the density—as in (2.24)–see also Remark 3. The additional input we offer is that the fluctuation of the charge density (or of a related operator) is precisely a very nontrivial condition of ODLRO ((2.32) or (4.37) respectively). As a final question, the treatment of the free Bose gas suggests that the chemical potential .μ(λ) < 0 for .λ /= 0 even after the thermodynamic limit also for interacting systems. It should be interesting to look at Bose gases with repulsive interactions [9] from the point of view of quasi-average: .(SSB)qa , using the symmetry breaking term (4.7).

Appendix In this Appendix we reproduce, for the reader’s convenience, the statement of the basic theorem of Fannes, Pulè and Verbeure [13], see also [29] for the extension to nonzero momentum, and Verbeure’s book [37]. Unfortunately, neither [13] nor [29] show that the states .ωβ,μ,φ , φ ∈ [0, 2π ) in the theorem below are ergodic. The simple, but instructive proof of this fact was given by Verbeure in his book [37], Theorem 4.5. Theorem 2 Let .ωβ,μ be an analytic, gauge-invariant equilibrium state. If .ωβ,μ exhibits ODLRO (4.37), then there exist ergodic states .ωβ,μ,φ , φ ∈ [0, 2π ), not gauge invariant, satisfying (1) .∀θ, φ ∈ [0, 2π ) such that .θ /= φ, .ωβ,μ,φ /= ωβ,μ,θ ; (2) the state .ωβ,μ has the decomposition ωβ,μ =

.

1 2π



2π

dφωβ,μ,φ . 0

(3) For each polynomial Q in the operators .η(b0 ),.η(b0∗ ), and for each .φ ∈ [0, 2π ), √ √ ωβ,μ,φ (Q(η(b0∗ ), η(b0 )X) = ωβ,μ,φ (Q( ρ0 exp(−iφ), ρ0 exp(iφ)X) ∀X ∈ A .

.

We remark, with Verbeure [37], that the proof of Theorem 2 is constructive. One essential ingredient is the separating character (or faithfulness) of the state .ωβ,μ , i.e., .ωβ,μ (A) = 0 implies .A = 0. This property, which depends on the '' extension of .ωβ,μ to the von-Neumann algebra .πω (A ) (see [11, 17]) is true for thermal states, but is not true for ground states, even without this extension: in fact, a ground state (or vacuum) is non-faithful on .A (see Proposition 3 of [39]). We

On Ergodic States

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see, therefore, that thermal states and ground states might differ with regard to the ergodic decomposition (ii). Compare also with our discussion in the Concluding remarks. Acknowledgments Some of the issues dealt with in this paper originate in the open problem posed in Sec.3 of [33] and at the of [18]. One of us (W.F.W.) would like to thank G. L. Sewell for sharing with him his views on ODLRO along several years. He would also like to thank the organisers of the Satellite conference “Operator Algebras and Quantum Physics” of the XVIII conference of the IAMP (Santiago de Chile) in São Paulo, July 17th-23rd 2015, for the opportunity to present a talk in which some of the ideas of the present paper were discussed. We are thankful to Bruno Nachtergaele for very useful remarks, suggestions, and corrections, which greatly improved and clarified the paper.

References 1. M.R. Andrews, C.G. Townsend, H.J. Miesner, D.S. Durfee, D.M. Kurn, W. Ketterle, Science 275, 637 (1997) 2. S. Attal, A. Joye, C.A. Pillet (eds.), Open quantum systems I. Lecture notes in mathematics, vol. 1880 (Springer Verlag, Berlin, 2006) 3. M. Beau, V.A. Zagrebnov, The second critical density and anisotropic generalised condensation. Condensed Matter Phys. 13, 23003, 1–10 (2010) 4. N.N. Bogoliubov, Lectures on quantum statistics: Quasi-Averages, vol. 2 (Gordon and Breach Science Publisher, Philadelphia, 1970) 5. N.N. Bogoliubov, Quasi-averages in problems of statistical mechanics Communication D781, pp.1–123, Laboratory of Theoretical Physics, JINR 1961. Republished in Collection of scientific works in twelve volumes. Statistical Mechanics, volume 6: Chapter II (Nauka, Moscow, 2006) 6. N.N. Bogoliubov, Collection of scientific works in twelve volumes. Statistical mechanics Theory of Nonideal Bose Gas, Superfluidity and Superconductivity, vol. 8 (Nauka, Moscow , 2007) 7. N.N. Bogoliubov, N.N. Bogoliubov Jr., Introduction to quantum statistical mechanics, 2nd edn. (World Scientific Publisher, Singapore, 2010) 8. M. van den Berg, J.T. Lewis, J.V. Pulè, A general theory of Bose Einstein condensation. Helv. Phys. Acta 59, 1271–1288 (1986) 9. O. Bratelli, D.W. Robinson, Equilibrium states of a Bose gas with repulsive interactions. Aust. J. Math. B 22, 129 (1980) 10. O. Bratteli, D.W. Robinson, Operator algebras and quantum statistical mechanics I (Springer Verlag, Berlin, 1987) 11. O. Bratteli, D.W. Robinson, Operator algebras and quantum statistical mechanics II, 2nd edn. (Springer Verlag, Berlin, 1997) 12. H.U.M. Domingos, F.W. Walter, Asymptotic time decay in quantum physics (World Scientific, Singapore, 2013) 13. M. Fannes, J.V. Pulè, A. Verbeure, On Bose condensation. Helv. Phys. Acta 55, 391–399 (1982) 14. J. Ginibre, On the asymptotic exactness of the Bogoliubov approximation for many boson systems. Commun. Math. Phys. 8, 26–51 (1968) 15. R.B. Griffiths, Spontaneous magnetization in idealized ferromagnets. Phys. Rev. 152, 240–246 (1966) 16. R. Haag, Local quantum physics - Fields, particles, algebras (Springer Verlag, Berlin, 1996)

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17. N.M. Hugenholtz, States and representations in statistical mechanics, in Mathematics of contemporary physics, ed. by R.F. Streater (Academic Press, New York, 1972) 18. T. Jaeck, V.A. Zagrebnov, Exactness of the Bogoliubov approximation in random external potentials. J. Math. Phys. 51, 123306, 1–16 (2010) 19. R.V. Kadison, States and representations. Trans. Am. Math. Soc. 103, 304–319 (1962) 20. N. Kawakami, M.C. Nemes, W.F. Wreszinski, The Casimir effect for parallel plates revisited. J. Math. Phys. 48, 102302 (2007) 21. E.H. Lieb, The classical limit of quantum spin systems. Commun. Math. Phys. 31, 327–340 (1973) 22. E.H. Lieb, F.J. Dyson, B. Simon, Phase transitions in quantum spin systems with isotropic and non-isotropic interactions. J. Stat. Phys. 18, 335–383 (1978) 23. E.H. Lieb, R. Seiringer, J. Yngvason, Justification of c-number substitutions in Bosonic Hamiltonians. Phys. Rev. Lett. 94, 080401:1–4 (2005) 24. E.H. Lieb, R. Seiringer, J. Yngvason, Bose Einstein condensation and spontaneous symmetry breaking. Rep. Math. Phys. 59, 389–399 (2007) 25. J.H. Lowenstein, J.A. Swieca, Quantum electrodynamics in two dimensions. Ann. Phys. 68, 172–195 (1971) 26. P.A. Martin, V.A. Zagrebnov, The Casimir effect for the Bose-gas in slabs. Europhys. Lett. 73, 15–20 (2006) 27. B. Nachtergaele, Quantum spin systems after DLS 1978. Proc. Symp. Pure Math. 76, 47–68 (2007) 28. J.V. Pulè, V.A. Zagrebnov, The canonical perfect Bose gas in Casimir boxes. J. Math. Phys. 45, 3565–3583 (2004) 29. J.V. Pulè, A. Verbeure, V. Zagrebnov, On nonhomogeneous Bose condensation. J. Math. Phys. 46, 083301, 1–8 (2005) 30. D. Ruelle, Statistical mechanics - rigorous results (W. A. Benjamin Inc., New York, 1969) 31. G.L. Sewell, Quantum theory of collective phenomena (Oxford University Press, Oxford, 1986) 32. G.L. Sewell, Quantum mechanics and its emergent macrophysics (Princeton University Press, Princeton, 2002) 33. G.L. Sewell, W.F. Wreszinski, On the mathematical theory of superfluidity. J. Phys. A Math. Theor. 42, 015207, 1–23 (2009) 34. B. Simon, The statistical mechanics of lattice gases, vol. 1 (Princeton University Press, Princeton, 1993) 35. A. Sütö, Equivalence of Bose Einstein condensation and symmetry breaking. Phys. Rev. Lett. 94, 080402 (2005) 36. J.A. Swieca, Goldstone theorem and related topics, in Cargese lectures in physics, vol. 4, ed. by D. Kastler (Gordon and Breach, New York, 1970) 37. A. Verbeure, Many body Boson systems - half a century later (Springer Verlag, Berlin, 2011) 38. W.F. Wreszinski, Charges and symmetries in quantum theories without locality. Fortschr. der Physik 35, 379–413 (1987) 39. W.F. Wreszinski, Passivity of ground states of quantum systems. Rev. Math. Phys. 17, 1–14 (2005) 40. W.F. Wreszinski, E. Abdalla, A precise formulation of the third law of thermodynamics. J. Stat. Phys. 134, 781–792 (2009) 41. W.F. Wreszinski, V.A. Zagrebnov, Bogoliubov quasi-averages: spontaneous symmetry breaking and the algebra fluctuations. Theor. Math. Phys. 194, 157–188 (2018) 42. V.A. Zagrebnov, The Bogoliubov c-Number Approximation for Random Boson Systems. Proc. Kiev Inst. Math. 11(1), 123–140 (2014) 43. V.A. Zagrebnov, Non-conventional dynamical Bose condensation. Physics of Particles and Nuclei 52, 202–238 (2021) 44. V.A. Zagrebnov, J.B. Bru, The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350, 291–434 (2001)

Improvement of the Hardy Inequality and Legendre Polynomials Nikolaos B. Zographopoulos

1 Introduction In this note we review some results from [20] concerning the improvement of the Hardy inequality on .RN and its relation with the Legendre polynomials. Hardy inequality states that the functional 

 I [φ] :=

.

RN

|∇φ|2 dx −

N −2 2

2  RN

φ2 dx , |x|2

(1)

is positive for any .φ ∈ C0∞ (RN ). Moreover, the coefficient .(N − 2)2 /4 is the best constant. For Hardy type inequalities and related topics we refer to [2, 5, 7–9, 11, 12]. We note that the Hardy inequality on .RN is sharp; there is no possibility to be improved by a norm. This means that the Hardy functional .I [φ] cannot be bigger than a (weighted or not) norm of .φ, for any .φ ∈ C0∞ (RN ). To see this consider the function e1 = |x|−(N −2)/2 ,

.

x ∈ RN /{0}.

which satisfies .

− Δe1 −

c∗ e1 = 0, |x|2

x ∈ RN /{0}.

(2)

Observe that .e1 /∈ L2 (RN ), because of the behavior at infinity.

N. B. Zographopoulos () Department of Mathematics and Engineering Sciences, University of Military Education, Hellenic Army Academy, Athens, Greece © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 N. J. Daras et al. (eds.), Exploring Mathematical Analysis, Approximation Theory, and Optimization, Springer Optimization and Its Applications 207, https://doi.org/10.1007/978-3-031-46487-4_18

459

460

N. B. Zographopoulos

However, in [20] it was proved that Theorem 1 For any .φ ∈ C0∞ (RN \{0}), holds that  I [φ] = I [φ ] ≥ CH P 1

.

RN

(1 + |x|2 )−2 |φ 1 |2 dx.

where .φ 1 is defined as the projection of .φ on the orthogonal complement of .e1 in L2 (RN , (1 + |x|2 )−2 dx).

.

For the general case for non smooth functions and/or with noncompact support (3) takes the form given in Proposition 1. The best constant and the minimizers of (3) are given in connection with the Legendre eigenvlaues λrn = 4n(n + 1) + 1 = (2n + 1)2 ,

.

n = 0, 1, . . .

(3)

and the associated eigenfunctions: Pn (t) =

.

 1 dn  2 n (t . − 1) 2n n! dt n

(4)

More presicely, Theorem 2 The best constant in (3) is CH P = λr2 − λr1 = 8,

.

independent of the dimension N. The best constant .CH P is not attained by a smooth function. However, it is attained, in the sense of Proposition 1, by the function e2 (x) = |x|

.

− N−2 2

 P2

|x|2 − 1 |x|2 + 1



= |x|−

N−2 2

|x|2 − 1 . |x|2 + 1

This function is the unique minimizer. Inequality (3) may also be seen as a natural way to bound from below the Hardy’s inequality in terms of the distance from the optimal function .e1 (see for instance the open problem in [1, pg. 75] stated for the Sobolev inequality). For a non standard improvement of the Hardy inequality we refer to [4], where also the function .e1 plays a crucial role. In [15] it was proved that .I [φ] maybe be improved with the 2 .L -norm of .φ, for .φ belonging to a certain class of functions. Finally we mention the recent works [3, 6, 13] dealing with the Hardy inequality on .RN .

Improvement of the Hardy Inequality and Legendre Polynomials

461

2 Proof of Theorem 1 In this Section we give a sketch of the proof of Theorem 1. For the details we refer to [20]. Denote by g any function that satisfies .(G ) g is a smooth, positive and radial function defined on the whole space. The behavior of g at infinity is .g(r) ∼ r ω , .r ↑ ∞, with .ω < −2 and .r = |x|.

2.1 Space Setting For g satisfying .(G ), we define the functional space .Hg as the completion of the C0∞ (RN )− functions under the norm

.

||φ||2Hg = I [φ] + ‖φ‖2L2 (RN ) ,

.

g

φ ∈ C0∞ (RN ).

(5)

Concerning the form of the norm, we may follow exactly the arguments of [14], which in turn are based on [16, 17]. Summarizing these results we have the following lemma: Lemma 1 The set .C0∞ (RN \{0}) is dense in .Hg , every function u, such that .||u||Hg < ∞, belongs to .Hg and the norm of .Hg is given by .

‖u‖2Hg := lim(IAε (u) − Λε (u) + Λ1/ε (u)) + ‖u‖2L2 (RN ) , ε↓0

g

(6)

where 

 IAε (u) :=

|∇u| dx − 2

.

Aε

N −2 2

2  Aε

|u|2 dx, |x|2

is the Hardy functional on the ring .Aε := B1/ε (0)−Bε (0), .B1/ε (0) and .Bε (0) denote the balls centered at the origin with radius .1/ε and .ε, respectively. The surface integrals N − 2 −1 .Λα (u) := α 2

 |x|=α

|u|2 dS,

represent the Hardy energy at the singularities. Proof Since g is a smooth bounded function we follow exactly the same steps as in [14, Section 2]. The proof is based on the transformation u(x) = |x|−

.

N−2 2

v(x),

(7)

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N. B. Zographopoulos

and the fact that the space .Hg with norm  ||v||2Hg =

.

RN

|x|−(N −2) |∇v|2 dx +

 RN

|x|−(N −2) g |v|2 dx.

(8) █

is equivalent to .Hg .

.

This energy formulation was first introduced in [16], for purpose of overcoming the functional difficulty emerged in the eigenfunctions that present the maximal singularity. Further applications of the consideration above, regarding singular parabolic problems, can be found in [14, 17]. A description of how the singularity energy .Λε (or .Λ1/ε ) behaves, maybe found at [16, Subsection 2.3]. Actually (5) implies that .

lim(IAε [u] − Λε (u) + Λ1/ε (u)) ≥ 0, ε↓0

u ∈ Hg ,

which maybe seen as an (optimal) generalization of the Hardy Inequality on .RN , for non smooth functions and/or for functions with noncompact support, (in the sense that they do not belong in the standard Sobolev spaces), see also [19]. Next, we do some comments about the selection of g. Assuming g to satisfy .(G ), we allow the effects from both singularities to be present. The function that plays a crucial role is .e1 (x) = |x|−(N −2)/2 , since (2) holds. On the other hand, assuming g to satisfy .(G ), we have that .e1 ∈ L2g (RN ). Observe that the Hardy functional .IAε [e1 ] vanishes: IAε [|x|−(N −2)/2 ] = 0.

(9)

.

We also have that Λε (e1 ) =

.

N − 2 −1 ε 2

 |x|=ε

|x|−N +2 dS =

N −2 , 2

and Λε−1 (e1 ) =

.

N −2 ε 2

 |x|=ε−1

|x|−N +2 dS =

N −2 , 2

i.e., the two singularity energies in (6) cancel each other. Thus, e1 ∈ Hg ,

.

and its norm is equal to  ||e1 ||2Hg = ||e1 ||2L2 (RN ) =

.

g

RN

g |x|−(N −2) dx.

(10)

Improvement of the Hardy Inequality and Legendre Polynomials

463

We also have that all functions behaving like .|x|−(N −2)/2 at infinity, belong to the space .Hg . Hence Lemma 2 The function .e1 = |x|−(N −2)/2 belongs in .Hg , satisfies .

lim(IAε [e1 ] − Λε (e1 ) + Λ1/ε (e1 )) = 0, ε↓0

and its norm is given by (10). Finally, we have the following imbedding. Lemma 3 Let g satisfy .(G ). Then, Hg ͨ→ L2g (RN ).

(11)

.

2.2 The Eigenvalue Problem In this subsection we study the eigenvalue problem .

c∗ u = (λ − 1) g u, |x|2

− Δu −

x ∈ RN \{0}.

(12)

Using the compact imbedding (11) we may prove the following result. Theorem 3 There exists an orthonormal basis .{ek }k≥1 of .L2g (RN ) constituted by eigenvectors of L with eigenvalue sequence 0 < λ1 ≤ λ2 ≤ . . . ≤ λk ≤ . . . → ∞.

.

(13)

so that .

− Δek −

c∗ ek + g u = λk g ek , |x|2

x ∈ RN \{0}.

Proof By construction, the distributional operator L given by L(u) = −Δu −

.

c∗ u + g u, |x|2

(14)

is the Riesz isomorphism from .Hg into its dual .Hg' and from (11) we have the triplet Hg → L2g (RN ) → Hg' .

.

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Then, by restriction to .L2g (RN ), we may define the surjective operator .L∗ : D(L∗ ) ⊂ L2g (RN ) → L2g (RN ), with domain   c∗ 2 N u + g u ∈ Lg (R ) . .D∗ = f ∈ Hg : −Δu − |x|2 If we denote .L∗ with L, is easy to see that L is self-adjoint with compact inverse. So, we can form an orthonormal basis of eigenfunctions in .L2g (RN ) as Theorem describes and the proof is completed. .█ Remark 1 Let .e˜1 be the normalization of .e1 in .L2g (RN ): e˜1 :=

.

RN

|x|−(N −2)/2 . |x|−(N −2)/2 g dx

In what follows, we use the same symbol .e1 both for the function .|x|−(N −2)/2 and for .e˜1 . We note that both sequences .{ek } and .{λk } depend on g. What is important and interesting is that we can write down explicitly the first eigenpair .(λ1 , e1 ) = (1, |x|−(N −2)/2 ), which is independent on g. Lemma 4 The principal eigenpair of (12) is (λ1 , e1 ) = (1, e˜1 ),

(15)

.

for any g satisfying .G . In addition, .λ1 is the unique eigenvalue of (12) with a positive associated eigenfunction and is simple. Proof Standard regularity results imply that if u is a weak solution of the problem (12), then .u ∈ C 2,ζ (RN \{0}), for some .ζ ∈ (0, 1). Since .λ1 is given by λ1 = inf

.

u∈Hg

||u||2H ||u||2L2 (RN )

,

g

we have that if .φ is an eigenfunction corresponding to .λ1 , then .|φ| is also an eigenfunction associated to .λ1 . Moreover, from Maximum principle we have that N 2 N .|φ(x)| > 0, .x ∈ R \{0}). The orthonormality of the eigenfunctions in .Lg (R ) implies the simplicity and the uniqueness up to positive eigenfunctions of .λ1 . On the other hand, (2) implies that .(1, |x|−(N −2)/2 ) is an eigenpair of (12). Thus (15) is true and the proof is completed. .█

Improvement of the Hardy Inequality and Legendre Polynomials

465

2.3 (Improved) Hardy-Poincaré Inequality Using the results of the previous sections, we may prove that the Hardy inequality on .RN maybe improved by a norm. More precisely, we have that Proposition 1 (Improved) Hardy-Poincaré inequality: For any .u ∈ Hg , holds that .

lim(IAε [u] − Λε (u) + Λ1/ε (u)) = lim(IAε [u1 ] − Λε (u1 ) + Λ1/ε (u1 )) ε↓0

ε↓0



≥ CH P (g)

RN

g |u1 |2 dx.

(16)

where .u1 ∈ X and .cg (u) ∈ R are defined through (17), (18). .CH P (g) is given by CH P (g) := λ2 − 1,

.

where .λ2 is the second eigenvalue of the problem (12) and is the best constant of the inequality. Proof Since .Hg ⊂ L2g (RN ) and the eigenfunctions .{ek } consist an orthonormal basis of .L2g (RN ), every function .u ∈ Hg maybe written as a sum u = cg (u) e1 + u1 = cg (u) |x|−(N −2)/2 + u1 ,

.

(17)

where cg (u) =

.

|x|−(N −2)/2 g u dx . −(N −2) g dx RN |x|

R N

(18)

and .u1 ∈ X, where X is the orthogonal complement of .e1 in .L2g (RN ) i.e. X = span{e2 , e3 , . . .}.

.

It is also clear that .u1 belongs also in .Hg . Step 1

We prove the equality in (16). Denoting by .K[u] the quantity K[u] = lim(IAε [u] − Λε (u) + Λ1/ε (u)),

.

ε↓0

we have to prove K[u] = K[u1 ].

.

(19)

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N. B. Zographopoulos

From (17) we get that K[u] = K[cg (u) e1 + u1 ] = cg2 K[e1 ] + K[u1 ] +

  1 +2cg ∇e1 · ∇u dx − c∗

.

Aε

−2cg

N −2 2

 e1 u1 dx 2 Aε |x|   e1 u1 dS + ε e1 u1 dS .

 1 ε ∂Bε

∂Bε−1

However, Lemma 2 implies that .K[e1 ] = 0 and integration by parts yields 

 ∇e1 · ∇u dx = c∗ 1

.

Aε

Aε

e1 u1 dx − |x 2 |

 ∂Bε

e1' (r) u1 dS

 +

∂Bε−1

e1' (r) u1 dS

where  .

− ∂Bε

e1' (r) u1 dS

N −21 = 2 ε

 e1 u1 dS, ∂Bε

and  .

e1' (r) u1 dS = −

∂Bε−1

N −2 ε 2

 ∂Bε−1

e1 u1 dS.

Finally, we conclude that (19) is true. Step 2 We prove the inequality in (16). The fact that .u1 belongs in X, according to the discussion in the previous Subsection, implies that ||u1 ||2Hg ≥ λ2 ||u1 ||2L2 (RN ) ,

.

g

where .λ2 > λ1 = 1 is the second eigenvalue of (12). This lead us to the inequality:  K[u1 ] ≥ (λ2 − 1)

.

RN

g |u1 |2 dx.

It is also clear that .CH P (g) = λ2 − 1 is the best constant of the inequality and the proof is competed. .█ Proof of Theorem 1 Assume that .φ belongs to .C0∞ (RN ). Then, according to (17), −(N −2)/2 + φ 1 , with .φ 1 ∈ X. Since .φ has a .φ may be written as .φ = cg (φ) |x| compact support, we have K[φ] = I [φ].

.

Improvement of the Hardy Inequality and Legendre Polynomials

467

Similarly, for .φ 1 holds that K[φ 1 ] = I [φ 1 ],

.

since the singularity terms in (16) cancel each other, as we saw in Lemma 2. The result now follows from Proposition 1. .█ The action of the Hardy functional I on a function .φ ∈ C0∞ (RN ), is nothing else than the action of I on .φ 1 , where .φ 1 is the projection of .φ in X. Note that .φ 1 is not a smooth function at the origin and does not have a compact support. Actually, 1 .φ has the critical behavior both at the origin and at infinity; its norm is understood through (6). Then, the Hardy functional is improved by the .L2g (RN ) norm of .φ 1 .

2.4 The Special Case g = (1 + r 2 )−2 In this Subsection we will calculate the radial sub-family of eigenfunctions of (12) and the associated eigenvalues. The reason is that we will try to give more information about the space X, the orthogonal complement of .e1 in .L2g (RN ) and as a result we will obtain the best constant and the minimizer in (16). Lemma 5 The set of eigenfunctions associated with the eigenvalue problem (12) contains two families of eigenfunctions: 1. The complete family of radial eigenfunctions en (r) = r −(N −2)/2 Pn

.



 r2 − 1 , r2 + 1

n = 0, 1, . . .

where .Pn are the Legendre polynomials given by (4). The associated eigenvalues are given by λrn = 4n(n + 1) + 1 = (2n + 1)2 ,

.

n = 0, 1, . . .

The family .{en (r)} consists a complete orthogonal basis of the radial subspace of .L2 (RN , (1 + r 2 )−2 dr). 2. A family of nonradial (i.e. with zero mean value) eigenfunctions √ cj

φn (r) = r −(N −2)/2+

.

(1 + r 2 )−cj fj (σ ), j = 1, 2, . . .

with associated eigenvalues λnr j = 4cj (cj + 1), j = 1, 2, . . . ,

.

468

N. B. Zographopoulos

where .fj (σ ) and .cj = j (j +N −2) denote the eigenfunctions and the associated eigenvalues, respectively, of the Laplace-Beltrami operator. Proof We are looking for eigenfunctions of the form e(r, σ ) = φ(r) fj (σ ).

.

Then .φ has to satisfy the eigenvalue equation N − 1 ' c∗ φ − 2 φ+ .−φ − r r ''



cj (λ − 1) − r2 (1 + r 2 )2

 φ = 0.

(20)

As a condition, we will assume that .φ ' ∈ L2 (R, r N −1 (1 + r 2 )−2 dx). Setting now ψ = r N −2 φ,

.

we conclude that .ψ solves   cj (λ − 1) 1 ' .ψ + ψ = 0. ψ + − 2+ r r (1 + r 2 )2 ''

(21)

A. Radial subfamily Looking for radial solutions, we set .j = 0, i.e. .cj = 0. Then (21) takes the form: 1 ' λ−1 ψ + ψ = 0. r (1 + r 2 )2

ψ '' +

.

(22)

Next we do the following change of variables: ψ(r) = w(t),

.

t=

r2 − 1 , r2 + 1

r > 0.

Note that .t ∈ (−1, 1) and the eigenvalue problem (22) is written as: d . dt

  λ−1 2 dw (1 − t ) + w = 0, dt 4

(23)

which is the Legendre differential equation. Since we want w to be regular both at .−1 and 1, we conclude that the eigenvalues of (23) are the Legendre polynomials (4), with corresponding eigenvalues .

λ−1 = n(n + 1), 4

and .Pn (t) is a complete orthogonal basis of .L2 (−1, 1).

Improvement of the Hardy Inequality and Legendre Polynomials

469

Looking for non-radial solutions, we set .j > 0, i.e. .j =

B. Non-radial subfamily 1, 2, . . .. Setting

ψ(t) = t

.

√ cj

h(t),

Eq. (21) takes the form: h'' +

.

2cj + 1 ' λ−1 h + h = 0. r (1 + r 2 )2

(24)

Assuming now that h(t) = (1 + t 2 )−cj ,

.

we conclude that 2 λnr j = (2cj + 1) − 1,

.

█

and the proof is completed.

.

Some Comments for the Set of the Radial Family of Eigenfunctions Let us first remind some results obtained in [18]. In the case of the unit ball .B1 centered at the origin (bounded domain case), the eigenvalue problem: .

− Δu −

c∗ u = λ u, |x|2

u|∂B1 = 0,

x ∈ B1 ,

(25)

admits a two parameter family of eigenfunctions

−(N −2)/2 .hj,n (r, σ ) = r Jm zm,n r fj (σ ), with free parameters .j ≥ 0, .n ≥ 1, .zm,n is the nth zero of the Bessel function .Jm and .m2 = cj = j (j + N − 2). The pair .(cj , fj ) is the j -eigenpair of the LaplaceBeltrami operator. The corresponding eigenvalues are 2 μj,n = zm,n .

.

The family .{hj,n } is a complete orthogonal basis of .L2 (B). As it is pointed in [18] the functions in H , the energetic space; see also [16], with zero radial part are smooth enough (in the sense that they belong in .H01 (B1 )). The maximal singularity is met only for radial functions; This maybe also be seen from the fact that the radial subspace of H has a complete basis of the form h0,n (r) = O(r −(N −2)/2 ).

.

In our case the radial eigenfunctions are related with the Legendre polynomials; The first eigenpair, as it was expected, is .(λ1 , e1 ) = (1, r −(N −2)/2 ). The second

470

N. B. Zographopoulos

eigenpair is given by r .(λ2 , e2 (r))

  r2 − 1 − N−2 2 . = 9, r r2 + 1

(26)

Then, the Bonnet’s recursion formula (n + 1) Pn+1 (x) = (2n + 1) x Pn (x) − n Pn−1 (x),

.

imply that also the next radial eigenfunctions have the maximal singularity both at the origin and at infinity. Moreover, the eigenvalues .λrn are independent of the dimension N . This is also the case for .B1 . Finally, it is immediate to say that in the radial case the best constant in (3) is equal to radial r CH P (g) = λ2 − 1 = 8,

.

and the unique associated eigenfunction is .e2 (r). Some Comments for the Set of the Nonradial Family of Eigenfunctions First we note that we are not in the position to prove that the nonradial eigenfunctions obtained in the previous Lemma consist the complete set of nonradial eigenfunctions. However, we may prove that every non radial function in .Hg is smooth enough (in the sense that they belong in .D 1,2 (RN )). We remind that the space .D 1,2 (RN ), .N ≥ 3, is defined as the closure of the .C0∞ (RN )- functions with respect to the norm  2 .||u|| 1,2 N D (R )

=

RN

|∇u|2 dx.

It can be shown that   2N D 1,2 (RN ) = u ∈ L N−2 (RN ) : |∇u| ∈ L2 (RN ) .

.

Equivalently, see for instance [10, Chapter 8], we may define .D 1,2 (RN ) as the set of 1 (RN )- functions f with .|∇f | ∈ L2 (RN ) and f vanishes at infinity. Moreover, .L loc there exists .K > 0 such that ||u||2D 1,2 (RN ) ≥ K ||u||22N ,

(27)

.

N−2

for all .u ∈ D 1,2 (RN ). The best constant in (27) is given by S(N) =

.

N (N − 2) |SN |2/N = 22/N π 1+1/N Γ 4



where .SN is the area of the N-dimensional unit sphere.

N +1 2

−2/N ,

(28)

Improvement of the Hardy Inequality and Legendre Polynomials

471

In addition, the imbedding D 1,2 (RN ) ͨ→ L2g (RN ),

(29)

.

is compact, for every .g ∈ LN/2 (RN ). To prove that the non radial functions of .Hg belong also to .D 1,2 (RN ), we argue in the same way as in [18]: Lemma 6 Every non radial function .unr ∈ Hg , belongs also to .D 1,2 (RN ). Proof Let .φnr be any non radial function belonging in .C0∞ (RN \{0}). Decompose the non radial function .φnr into spherical harmonics φnr =

∞ 

.

φj fj (σ ),

j =1

with .fj and .cj as in Lemma 5. Then I [φnr ] =



∞  

|∇φj | − (c∗ − cj )

φj2

2

.

N j =1 R

|x|2

 dx,

and Hardy’s inequality implies that I [φnr ] ≥

.

cj c∗

 |∇φnr |2 dx ≥

RN

N −1 c∗

 RN

|∇φnr |2 dx,

(30)

or  I [φnr ] ≥ (N − 1)

.

RN

|φnr |2 dx. |x|2

(31)

Then by completion we get that for every non radial function .unr ∈ Hg , we have that |∇unr | ∈ L2 (RN ),

.

(32)

and  .

RN

|unr |2 dx < ∞. |x|2

Last inequality implies that .unr behaves at the origin (at most) like .|x|−(N −2)/2 h(x), where h tends to zero at infinity. Thus, .unr tends to zero at infinity and finally we conclude that .unr ∈ D 1,2 (RN ). .█

472

N. B. Zographopoulos

Proof of Theorem 2 It suffices to prove that all non radial eigenvalues of (12) have associated eigenvalues bigger than .λr2 − 1 = 8. Since we are not in the position to prove that the nonradial eigenfunctions obtained in Lemma 5 consist the complete set of nonradial eigenfunctions, we use an indirect proof. Let .(λnr , unr ) be an eigenpair of (12), where .unr is non radial. From (30) we have that  N −1 |∇unr |2 dx. .I [unr ] ≥ N c∗ R Observe now that .g = (1 + r 2 )−2 is an .LN/2 (RN )- function. Thus, from (29), there exists a positive real number .ν1 , such that 



.

RN

|∇u|2 dx ≥ ν1

RN

g |u|2 dx,

for any .u ∈ D 1,2 (RN ). Then, we get that N −1 ν1 c∗

I [unr ] >

.

 RN

g |unr |2 dx.

Next we calculate .ν1 , which actually is the principal eigenvalue of the problem .

− Δu = ν g u,

x ∈ RN .

(33)

Using the same arguments as in Lemma 4 we have that .ν1 is simple and is the only eigenvalue which admits a positive eigenfunction. However, the function .(1 + |x|2 )−(N −2)/2 solves (33) with .ν = N(N − 2). Thus, ν1 = N(N − 2).

.

Since .

N −1 4N(N − 1) > 8, ν1 = N −2 c∗

we have that the best constant of (3) is .CH P = 8.

for every N, █

.

Corollary 1 Is clear from the above proof that the function .e2 is the unique minimizer of the Improved Hardy-Poincaré inequality, according to Proposition 1, for .g = (1 + r 2 )−2 .

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473

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