Progress in Approximation Theory: An International Perspective

462 Pages·1992·12.6 MB·English

by  N. M. Atakishiyev, S. K. Suslov (auth.), A. A. Gonchar, E. B. Saff (eds.)

Springer Series in
Computational
Mathematics
19
Editorial Board
R.L. Graham, Murray Hill (NJ)
J. Stoer, Warzburg
R. Varga, Kent (Ohio)
A.A. Gonchar E.B. Saff
Editors
Progress in
Approximation Theory
An International Perspective
Springer-Verlag
New York Berlin Heidelberg London Paris
Toyko Hong Kong Barcelona Budapest
AA Gonchar E.B. Saff
Steklov Mathematics Institute Institute for Constructive Mathematics
117966 Moscow GSP-1 University of South Florida
Russia Tampa, FL 33620
USA
Mathematics Subject Classification (1991): 30-06, 33-06, 41-06
With 9 figures.
Library of Congress Cataloging-in-Publication Data
Progress in approximation theory: an international perspectivel
edited by A.A. Gonchar and E.B. Saff.
p. cm. - (Springer series in computational mathematics;
19.)
"Proceedings of an international conference on approximation
theory that was held March 19-22, 1990, at the University of South
Florida, Tampa"-Pref.
Includes bibliographical references (p. 33-35) and index.
ISBN-13:978-1-4612-7737-8
1. Approximation theory-Congresses. I. Gonchar, A.A. (Andrei
A.) II. Saff, E.B., 1944- . III. Series.
QA221.P78 1992
511 '.4-dc20 92-24316
Printed on acid-free paper.
© 1992 Springer-Verlag New York, Inc.
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Preface
This volume is the proceedings of an international conference on approxi-
mation theory that was held March 19 to 22, 1990, at the University of
South Florida, Thmpa. The conference was a historic event in the sense that
it brought together for the first time a large number of approximation
theorists from the United States and from Russia, the Ukraine, and several
other former Soviet countries. In addition, 69 other delegates from Europe-
an, North American, and Asian countries were present.
The conference was hosted by the Institute for Constructive Mathemat-
ics, and the organizing committee consisted of A.A. Gonchar, E.B. Saff, S.
Khrushchev, R.A. DeVore, and P. Nevai. Both plenary and shorter invited
research announcements were presented.
This proceedings differs in several important respects from typical confer-
ence publications. First, all of the contributions that appear were by invita-
tion of the editors. Second, much latitude was granted to authors in prepar-
ing their manuscripts; the typical page limitations were relaxed in order to
encourage contributions that present not only new results but include de-
tailed perspectives on the subject area.
This volume is designed to give an overview of current research activities
in approximation theory and special functions that truly reflects the interna-
tional nature of these subject areas. For example, 8 of the 19 chapters are
authored or coauthored by mathematicians from former Soviet countries.
The contents include a wide range of topics, which we now briefly describe.
Of interest to mathematicians in special functions are new results dealing
with q-hypergeometric functions, difference hypergeometric functions, and
basic hypergeometric series with Schur function argument. Several chapters
concern the theory of orthogonal polynomials and expansions, including
generalizations of Szego type asymptotics and connections with Jacobi ma-
trices. The convergence theory for Pade and Hermite-Pade approximants is
explored in three chapters in which techniques from potential theory are
emphasized. The relatively new topics of wavelets and fractals are featured
in chapters dealing with invariant measures and nonlinear approximation.
Applications of results concerning approximation by entire functions and
the problem of analytic continuation are also included. In addition, gener-
alizations of de Brange's inequality for univalent functions are presented
from an operator point of view in a quasi-orthogonal Hilbert space setting.
v
vi Preface
Further topics include rearrangements of functions, harmonic analysis, nu-
merical estimates for the de Bruijn-Newman constant, and approximation
by polynomials with varying weights.
It is the sincere hope of the organizers that the conference along with this
proceedings will act as a catalyst for future joint research and extensive
cooperation among approximation theorists of all nations.
The organizers are particularly grateful for the support of the National
Science Foundation, the former Soviet Academy of Sciences, and the Uni-
versity of South Florida.
We are also indebted to the local organizing committee consisting of
M.E.H. Ismail, M. Parrott, B. Shekhtman, V. Thtik, and C. Williams for
their help in planning activities and hosting our foreign guests. Special notes
of appreciation are further extended to Maria Carvalho, who served effi-
ciently and energetically as the conference secretary, and to Rafael Munoz,
who carefully typed many of the manuscripts.
Moscow, Russia A.A. Gonchar
Thmpa, Florida E.B. Saf!
Contents
Preface..................................................... v
List of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Difference Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
N.M. Atakishiyev and S.K. Suslov
Pad~ Approximants for Some q-Hypergeometric Functions. . . . . . . . . 37
M.E.H. Ismail, R. Pertine, and J. Wimp
Summation Theorems for Basic Hypergeometric Series of
Schur Function Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
S.C. Milne
Orthogonal Polynomials, Recurrences, Jacobi Matrices,
and Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
P. Nevai
Szego lYPe Asymptotics for Minimal Blaschke Products. . . . . . . . . . . . 105
A.L. Levin and E.B. Sa/I
Asymptotics of Hermite-Pad~ Polynomials. . . . . . . . . . . . . . . . . . . . . . . 127
A.I. Aptekarev and H. Stahl
On the Rate of Convergence of Pad~ Approximants of
Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.A. Gonchar, E.A. Rakhmanov, and S.p. Suetin
Spurious Poles in Diagonal Rational Approximation. . . . . . . . . . . . . . . 191
D.S. Lubinsky
Expansions for Integrals Relative to Invariant Measures
Determined by Contractive Affine Maps......................... 215
C.A. Micchelli
Approximation of Measures by Fractal Generation Thchniques ...... 241
S.Demko
Nonlinear Wavelet Approximation in the Space C (Rd) ••••••••••••• 261
R.A. DeVore, P. Petrushev, and X.M. Yu
vii
viii Contents
Completeness of Systems of Translates and Uniqueness Theorems
for Asymptotically Holomorphic Functions . . . . . . . . . . . . . . . . . . . . .. 285
A.A. Borichev
Approximation by Entire Functions and Analytic Continuation ..... 295
N. U Arakelyan
Quasi-Orthogonal Hilbert Space Decompositions and Estimates
of Univalent Functions. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315
N.K. Nikolskii and V.I. Vasyunin
On the Differential Properties of the Rearrangements of Functions. .. 333
v.l. Kolyada
A Class of I.M. Vinogradov's Series and Its Applications
in Harmonic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 353
K.l. Oskolkov
A Lower Bound for the de Bruijn-Newman Constant A. II.......... 403
T.S. Norfolk, A. Ruttan, and R.S. Varga
On the Denseness of Weighted Incomplete Approximations . . . . . . . .. 419
P. Borwein and E.B. Saff
Asymptotics of Weighted Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .. 431
M. v. Golitschek, G. G. Lorentz, and Y. Makovoz
List of Participants
Dan Amir, Faculty of Exact Sciences, leI-Aviv University, Ramat Aviv
69978, Israel.
G.A. Anastassiou, Department of Mathematical Sciences, Memphis State
University, Memphis, TN 38152, USA.
Milne Anderson, Department of Mathematics, University College, London
WC16 EBT, UK.
A.I. Aptekarev, Keldysh Institute of Applied Mathematics, Miusskaya Sq.
4, 125047 Moscow A-47, Russia.
Norair U Arakelyan, Institute of Mathematics, Armenian Academy of Sci-
ences, M~shal Bagramian Ave. 24 - B, 375019 - Yerevan, Armenia.
Remi Arcangeli, Laboratoire de Mathematiques Appliquees, Universite de
Pau, Avenue de l'Universite, 64000 Pau, France.
Richard A. Askey, Department of Mathematics, University of Wisconsin,
Madison, WI 53706, USA.
Natig Mamed ogly Atakishiyev, Physics Institute, Narimanov pro 33, Baku
370143, Azerbaijan.
R V. Atkinson, Department of Mathematics, University of Toronto, Toron-
to, Canada M5S IA1.
Bogdan M. Baishanski, Department of Mathematics, Ohio State University,
Columbus, OH 43210-1174, USA.
George A. Baker, Jr., T-I1, MS-B262, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA.
Laurent Baratchart, Res. des Hartes, 700 ch. des Combes, 06600 Antibe,
France.
V.I. Belyi, Institute for Applied Mathematics and Mechanics, Ukrainian
Academy of Science, ul. Roze Luxemburg 74, Ukraine.
Hubert Berens, Mathematisches Institut, UniversiHit Erlangen - Niirnberg,
Bismarckstrasse 1112, D-8520 Erlangen, Germany.
ix
x Participants
Hans-Peter Blatt, Lehrstuhl fUr Mathematik-Angewandte Mathematik,
Katholische Univ. Eischstatt, Ostenstrasse 18, D-8078 Eichstatt, Germany.
Andre Boivin, Department of Mathematics, University of Western Ontario,
London, Canada N6A 5B7.
Ranko Bojanic, Department of Mathematics, Ohio State University, Co-
lumbus, OH 43210-1174, USA.
A.A. Borichev, Steklov Mathematical Institute, St. Petersburg Branch, 27
Fontanka, 191011 St. Petersburg, Russia.
Peter Borwein, Department of Mathematics, Dalhousie University, Halifax,
Canada B3H 4H8.
Dietrich Braess, Mathematisches Institut, Ruhr- Universitat, 4630 Bo-
chum, Germany.
Bruno Brosowski, Universitat Frankfurt, Fachbereich Mathematik, D-6000
Frankfurt a.M., Germany.
M.D. Buhmann, DAMTP, University of Cambridge, Silver Street, Cam-
bridge CB3 9EW, UK.
Paul L. Butzer, Lehrstuhl A. fUr Mathematik, Thchnishe Hochschule Aa-
chen, D-5100 Aachen, Germany.
Alfred Cavaretta, Jr., Mathematics Department, Kent State University,
Kent, OH 44242, USA.
Bruce L. Chalmers, Department of Mathematics, University of California,
Riverside, CA 92521, USA.
Jairo A. Charris, Department of Mathematics and Statistics, OF. 315, Na-
tional University of Colombia, Bogota, Colombia.
Weiyu Chen, Department of Mathematics, University of Alberta, Edmon-
ton, Canada T6G 2Gl.
Sandra Cooper, Department of Pure and Applied Mathematics, Washing-
ton State University, Pullman, WA 99163-2930, USA.
Carl de Boor, Department of Mathematics, University of Wisconsin, Madi-
son, WI 53705, USA.
Marcel G. de Bruin, Delft University of Thchnology, Faculty of Thchnical
Mathematics and Informatics, 2600 A J Delft, The Netherlands.
Jesus S. Dehesa, Departamento de Fisica Moderna, Facultad de Ciencias,
Universidad de Granada, 18071 Granada, Spain.
Participants xi
Biancamaria della Vecchia, Istituto per Applicazioni della Matematica-
CNR, 80131 Napoli, Italy.
Stephen Demko, School of Mathematics, Georgia Institute of Technology,
Atlanta, GA 30332, USA.
Baiquio Deng, Department of Mathematics, University of South Carolina,
Columbia, SC 29208, USA.
Catherin Detail/e, Department of Mathematics, Facultes Univ. ND de la
Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.
Ron A. De Yore, Department of Mathematics, University of South Carolina,
Columbia, SC 29208, USA.
z. Ditzian, Department of Mathematics, University of Alberta, Edmonton,
Alberta, Canada T6G 2G 1.
Andre Draux, Laboratoire d'Analyse Numerique, Universite de Lille, 59655
Villeneuve d'Ascq Cedex, France.
Michael Eiermann, Institiit fUr Praktische Mathematik, Universitiit Karls-
ruhe, D-7500 Karlsruhe 1, Germany.
Tamas Erdelyi, Department of Mathematics, Ohio State University, Colum-
bus, OH 43210-1174, USA.
Mosche Feder, Department of Mathematics, St. Lawrence University, Can-
ton, NY 13617, USA.
Bernd Fischer, Institute of Applied Mathematics, University of Hamburg,
D-2000 Hamburg 13, Germany.
Thomas Fischer, Fachbereich Mathematik, Wolfgang Goethe-Universtitiit,
6000 Frankfurt 1, Germany.
Wolfgang H. Fuchs, Department of Mathematics, Cornell University, Itha-
ca, NY 14853, USA.
Dieter Gaier, Department of Mathematics, University of Giessen, Arndt-
strasse 2, 63 Giessen, Germany.
Jeffrey Geronimo, School of Mathematics, Georgia Institute of Technology,
Atlanta, GA 30332, USA.
Manfred v. Golitschek, Institiit fUr Angewandte Mathematik, Universitiit
Wiirzburg, Am Hubland, 8700 Wiirzburg, Germany.
Andrei A. Gonchar, Steklov Mathematics Institute, Vavilova 42, 117966
Moscow GSP-l, Russia.