Springer Monographs in Mathematics Amnon Jakimovski, Ambikeshwar Sharma and József Szabados Walsh Equiconvergence of Complex Interpolating Polynomials BA C Amnon Jakimovski Ambikeshwar Sharma Tel-Aviv University The University of Alberta Tel-Aviv, Israel Edmonton, Canada József Szabados Hungarian Academy of Sciences Budapest, Hungary A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-4174- 7 (HB) ISBN 978-1-4020-4175- 4 (e-Book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved ⃝c 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. DEDICATION And one might therefore say of me that in this book, I have only made up a bunch of other people’s flowers and that of my own I have only provided the string that ties them together. (Book III, Chapter XVI of Physiognomy) Signeur de Montaigne v CONTENTS Preface xi 1. LAGRANGE INTERPOLATION AND WALSH EQUICONVERGENCE 1 1.1 Introduction 1 1.2 Least-Square Minimization 5 1.3 Functions Analytic in Γρ = {z : |z| = ρ} 7 1.4 An Extension of Walsh’s Theorem 11 1.5 Multivariate Extensions of Walsh’s Theorem 14 1.6 Historical Remarks 21 2. HERMITE AND HERMITE-BIRKHOFF INTERPOLATION AND WALSH EQUICONVERGENCE 25 2.1 Hermite Interpolation 25 2.2 Generalization of Theorem 1 30 2.3 Mixed Hermite Interpolation 33 2.4 Mixed Hermite and ℓ2-Approximation 38 2.5 A Lemma and its Applications 41 2.6 Birkhoff Interpolation 47 2.7 Historical Remarks 53 3. A GENERALIZATION OF THE TAYLOR SERIES TO RATIONAL FUNCTIONS AND WALSH EQUICONVERGENCE 55 3.1 Rational Functions with a Minimizing Property 55 n n 3.2 Interpolation on roots of z − σ 58 α 3.3 Equiconvergence of R n+m,n(z) and rn+m, n,(z) fo r m ≥ −1 61 3.4 Hermite Interpolation 69 3.5 A Discrete Analogue of Theorem 1 72 vii viii WALSH EQUICONVERGENCE OF COMPLEX INTERPOLATING . . . 3.6 Historical Remarks 80 4. SHARPNESS RESULTS 81 4.1 Lagrange Interpolation 81 4.2 Hermite Interpolation 95 4.3 The Distinguished Role of the Roots of Unity for the Circle 97 4.4 Equiconvergence of Hermite Interpolation on Concentric Circles 103 4.5 (0,m)-P´al type Interpolation 108 4.6 Historical Remarks 111 5. CONVERSE RESULTS 115 5.1 Lagrange Interpolation 115 5.2 Hermite Interpolation 124 5.3 Historical Remarks 128 ´ 6. PADE APPROXIMATION AND WALSH EQUICONVERGENCE FOR MEROMORPHIC FUNCTIONS WITH ν−POLES 129 6.1 Introduction 129 6.2 A Generalization of Theorem 1 132 6.3 Historical Remarks 142 7. QUANTITATIVE RESULTS IN THE EQUICONVERGENCE OF APPROXIMATION OF MEROMORPHIC FUNCTIONS 149 7.1 The main Theorems 149 7.2 Some Lemmas 150 7.3 Distinguished Points for |z| < ρ (proof of Theorems 1 and 2) 157 7.4 Distinguished Points for |z| ≥ ρ (proof of Theorem 3) 160 7.5 A Lemma and Proof of Theorem 4 166 7.6 Simultaneous Hermite-Pad´e Interpolation 173 7.7 Historical Remarks 175 8. EQUICONVERGENCE FOR FUNCTIONS ANALYTIC IN AN ELLIPSE 177 8.1 Introduction 177 8.2 Equiconvergence (Lagrange Interpolation 181 8.3 Equiconvergence (Hermite Interpolation) 186 8.4 Historical Remarks 191 9. WALSH EQUICONVERGENCE THEOREMS FOR THE FABER SERIES 199 9.1 Introducing Faber polynomials and Faber expansions 199 CONTENTS ix 9.2 Extended equiconvergence theorems for Faber expansions 205 9.3 Additional properties of Faber polynomials 208 9.4 Estimates of the polynomials ωn(z) for Fej´er and Faber nodes 211 9.5 Integral representations of Lagrange and Hermite interpolants for Faber expansions 214 9.6 Proofs of the theorems stated in Section 9.2 216 9.7 Historical Remarks 240 10. EQUICONVERGENCE ON LEMNISCATES 243 10.1 Equiconvergence on Lemniscates 243 10.2 Historical Remarks 255 11. WALSH EQUICONVERGENCE AND EQUISUMMABILITY 257 11.1 Introduction 257 α,β 11.2 Definition of the kernels (Λ n (z, f))n≥1 259 11.3 Equiconvergence and Equisummability of the operators α,β Λ (z, f) 262 n 11.4 Proof of Theorem 3.3 263 11.5 Some Topological Results 271 11.6 Proof of Theorem 3.4 282 11.7 Applications of Theorems 3.3 and 3.4 286 11.8 Historical Remarks 290 REFERENCES 291