Walsh Equiconvergence of Complex Interpolating Polynomials
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Amnon Jakimovski, Ambikeshwar Sharma and József Szabados
Walsh Equiconvergence of Complex Interpolating Polynomials
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Amnon Jakimovski Tel-Aviv University Tel-Aviv, Israel
Ambikeshwar Sharma The University of Alberta 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
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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
xi
Preface 1. LAGRANGE INTERPOLATION AND WALSH EQUICONVERGENCE 1.1 Introduction 1.2 Least-Square Minimization 1.3 Functions Analytic in Γρ = {z : |z| = ρ} 1.4 An Extension of Walsh’s Theorem 1.5 Multivariate Extensions of Walsh’s Theorem 1.6 Historical Remarks
1 1 5 7 11 14 21
2. HERMITE AND HERMITE-BIRKHOFF INTERPOLATION AND WALSH EQUICONVERGENCE 2.1 Hermite Interpolation 2.2 Generalization of Theorem 1 2.3 Mixed Hermite Interpolation 2.4 Mixed Hermite and 2 -Approximation 2.5 A Lemma and its Applications 2.6 Birkhoff Interpolation 2.7 Historical Remarks
25 25 30 33 38 41 47 53
3. A GENERALIZATION OF THE TAYLOR SERIES TO RATIONAL FUNCTIONS AND WALSH EQUICONVERGENCE 3.1 Rational Functions with a Minimizing Property 3.2 Interpolation on roots of z n − σ n α (z) and rn+m, n,(z) for m ≥ − 1 3.3 Equiconvergence of Rn+m,n 3.4 Hermite Interpolation 3.5 A Discrete Analogue of Theorem 1 vii
55 55 58 61 69 72
v iii
WALSH EQUICONVERGENCE OF COMPLEX INTERPOLATING . . .
3.6 Historical Remarks
80
4. SHARPNESS RESULTS 4.1 Lagrange Interpolation 4.2 Hermite Interpolation 4.3 The Distinguished Role of the Roots of Unity for the Circle 4.4 Equiconvergence of Hermite Interpolation on Concentric Circles 4.5 (0, m)-P´al type Interpolation 4.6 Historical Remarks
81 81 95 97 103 108 111
5. CONVERSE RESULTS 5.1 Lagrange Interpolation 5.2 Hermite Interpolation 5.3 Historical Remarks
115 115 124 128
´ APPROXIMATION AND WALSH EQUICONVERGENCE 6. PADE FOR MEROMORPHIC FUNCTIONS WITH ν−POLES 6.1 Introduction 6.2 A Generalization of Theorem 1 6.3 Historical Remarks
129 129 132 142
7. QUANTITATIVE RESULTS IN THE EQUICONVERGENCE OF APPROXIMATION OF MEROMORPHIC FUNCTIONS 7.1 The main Theorems 7.2 Some Lemmas 7.3 Distinguished Points for |z| < ρ (proof of Theorems
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