Frontiers in interpolation and approximation: dedicated to the memory of Ambikeshwar Sharma

476 Pages·2007·6.03 MB·English

by  N. K. Govil

Frontiers in Interpolation
and Approximation
Dedicated to the memory of Ambikeshwar Sharma

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Frontiers in Interpolation
and Approximation
Dedicated to the memory of Ambikeshwar Sharma
N. K. Govil
Auburn Univesity
Alabama, U.S.A.
H. N. Mhaskar
California State University
Los Angeles, U.S.A.
Ram N. Mohapatra
University of Central Florida
Orlando, U.S.A.
Zuhair Nashed
University of Central Florida
Orlando, U.S.A.
J. Szabados
Alfréd Rényi Institute of Mathematics
Budapest, Hungary
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Library of Congress Cataloging-in-Publication Data
Frontiers in interpolation and approximation / [edited by] N.K. Govil … [et al.].
p. cm. -- (Monographs and textbooks in pure and applied mathematics ; 282)
Includes bibliographical references and index.
ISBN-13: 978-1-58488-636-5 (acid-free paper)
ISBN-10: 1-58488-636-6 (acid-free paper)
1. Interpolation. 2. Approximation theory. I. Govil, N. K. (Narendra Kumar) II.
Series.
QA281.F76 2007
511’.42--dc22 2006013708
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Contents
Foreword ix
R. S. Varga
Preface xi
Editors xv
Contributors xvii
Ambikeshwar Sharma xxiii
A. Jakimovski and J. Szabados
1. Markov-Type Inequalities for Homogeneous Polynomials on 1
Nonsymmetric Star-Like Domains
Yuliya Babenko and Andr´as Kro´o
2. Local Inequalities for Multivariate Polynomials and 17
Plurisubharmonic Functions
A. Brudnyi and Yu. Brudnyi
3. The Norm of an Interpolation Operator on H∞(D) 33
A. S. Cavaretta and N. Fontes-Merz
4. Sharma and Interpolation, 1993–2003: The Dutch Connection 45
M. G. de Bruin
5. Freeness of Spline Modules from a Divided to a Subdivided 59
Domain
Satya Deo and J. K. Maitra
6. Measures of Smoothness on the Sphere 75
Z. Ditzian
7. Quadrature Formulae of Maximal Trigonometric Degree of 93
Precision
Dimiter Dryanov
8. Inequalities for Exponential Sums via Interpolation and 119
Tur´an-Type Reverse Markov Inequalities
T. Erd´elyi
9. Asymptotic Optimality in Time-Frequency Localization of 145
Scaling Functions and Wavelets
T. N. T. Goodman and S. L. Lee
10. Interpolation by Polynomials and Transcendental Entire 173
Functions
N. K. Govil, M. A. Qazi, and Q. I. Rahman
vii
11. Hyperinterpolation on the Sphere 213
Kerstin Hesse and Ian H. Sloan
12. Lagrange Interpolation at Lacunary Roots of Unity 249
A. Jakimovski
13. A Fast Algorithm for Spherical Basis Approximation 259
J. Keiner and J. Prestin
14. Direct and Converse Polynomial Approximation Theorems on 287
the Real Line with Weights Having Zeros
G. Mastroianni and J. Szabados
15. Fourier Sums and Lagrange Interpolation on (0,+∞) and 307
(−∞,+∞)
G. Mastroianni and P. V´ertesi
16. On Bounded Interpolatory and Quasi–Interpolatory 345
Polynomial Operators
H. N. Mhaskar
17. Hausdorff Strong Uniqueness in Simultaneous Approximation. 365
Part II
Devidas Pai and Indira K
18. Zeros of Polynomials Given as an Orthogonal Expansion 381
Gerhard Schmeisser
19. Uniqueness of Tchebycheff Spaces and Their Ideal Relatives 407
Boris Shekhtman
Index 427
viii
Foreword
Iamindeedhonoredtowritethisforewordforthispresentvolume
which is dedicated to Ambikeshwar Sharma, who was well known in
the world as a beloved teacher and research mathematician. He was
my dear and trusted friend and colleague for many, many years; we
first met, some fifty years ago, in the office of Professor Joseph L.
Walsh, my adviser at Harvard University. Little did I know how
much our lives would intertwine over the years.
Ambikeshwar had two well-recognized strengths which were
absolutely infectious to those near him. He loved mathematics and
he loved mathematical research with others, both with great vigor. I
can vividly recall working with him on some math paper (we wrote
13 papers together) when, at night, I would be exhausted, and he al-
ways was ready to continue onward! (I often wondered how much of
this came from his excellent and strict vegetarian diet.) His research
work was basically in complex function theory and approximation
theory, and he is probably best known for his work on splines, in-
terpolation theory, and Walsh over-convergence. On this last topic,
Walshover-convergence,itwashisdreamtowritethefirstanddefini-
tive book on this topic, which would examine in detail all aspects of
this theory. Unfortunately, his health gave out before this book was
finished, and it was left to his circle of friends in mathematics to
complete this task.
This book, which follows, is dedicated to Ambikeshwar Sharma,
whowilllongberememberedforhismathematics, forhisenthusiasm
for mathematical research, and for the overwhelming kindness and
understanding he showered on all who came in contact with him.
Richard S. Varga
Kent State University
ix