Frontiers in Interpolation and Approximation Dedicated to the memory of Ambikeshwar Sharma PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida Piscataway, New Jersey Orlando, Florida EDITORIAL BOARD M. S. Baouendi Freddy van Oystaeyen University of California, University of Antwerp, San Diego Belgium Jane Cronin Donald Passman Rutgers University University of Wisconsin, Madison Jack K. Hale Georgia Institute of Technology Fred S. Roberts Rutgers University S. Kobayashi University of California, David L. Russell Berkeley Virginia Polytechnic Institute and State University Marvin Marcus University of California, Walter Schempp Santa Barbara Universität Siegen W. S. Massey Mark Teply Yale University University of Wisconsin, Milwaukee Anil Nerode Cornell University MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles G. S. Ladde and M. Sambandham, Stochastic versus Deterministic Systems of Differential Equations (2004) B. J. Gardner and R. Wiegandt, Radical Theory of Rings (2004) J. Haluska, The Mathematical Theory of Tone Systems (2004) C. Menini and F. Van Oystaeyen, Abstract Algebra: A Comprehensive Treatment (2004) E. Hansen and G. W. Walster, Global Optimization Using Interval Analysis, Second Edition, Revised and Expanded (2004) M. M. Rao, Measure Theory and Integration, Second Edition, Revised and Expanded (2004) W. J. Wickless, A First Graduate Course in Abstract Algebra (2004) R. P. Agarwal, M. Bohner, and W-T Li, Nonoscillation and Oscillation Theory for Functional Differential Equations (2004) J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (2004) Walter Ferrer and Alvaro Rittatore, Actions and Invariants of Algebraic Groups (2005) Christof Eck, Jiri Jarusek, and Miroslav Krbec, Unilateral Contact Problems: Variational Methods and Existence Theorems (2005) M. M. Rao, Conditional Measures and Applications, Second Edition (2005) A. B. Kharazishvili, Strange Functions in Real Analysis, Second Edition (2006) Vincenzo Ancona and Bernard Gaveau, Differential Forms on Singular Varieties: De Rham and Hodge Theory Simplified (2005) Santiago Alves Tavares, Generation of Multivariate Hermite Interpolating Polynomials (2005) Sergio Macías, Topics on Continua (2005) Mircea Sofonea, Weimin Han, and Meir Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage (2006) Marwan Moubachir and Jean-Paul Zolésio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions (2006) Alfred Geroldinger and Franz Halter-Koch, Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory (2006) Kevin J. Hastings, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition (2006) Robert Carlson, A Concrete Introduction to Real Analysis (2006) John Dauns and Yiqiang Zhou, Classes of Modules (2006) N. K. Govil, H. N. Mhaskar, Ram N. Mohapatra, Zuhair Nashed, and J. Szabados, Frontiers in Interpolation and Approximation (2006) 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 Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-636-6 (Hardcover) International Standard Book Number-13: 978-1-58488-636-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. 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 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com T&F_LOC_G_Master.indd 1 6/1/06 3:28:23 PM 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