Functional Analysis and Approximation: Proceedings of the Conference held at the Mathematical Research Institute at Oberwolfach, Black Forest, August 9–16, 1980

460 Pages·1981·10.34 MB·English

by  P. L. Butzer

ISNM60:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur Numerischen Mathematik
Serie internationale d' Analyse numerique
Vol. 60
Edited by
Ch. Blanc, Lausanne; A. Ghizzetti, Roma;
R. Glowinski, Paris; G. Golub, Stanford;
P. Henrici, Zurich; H. O. Kreiss, Pasadena;
A. Ostrowski, Montagnola; J. Todd, Pasadena
Birkhauser Verlag
Basel· Boston· Stuttgart
Functional Analysis
and Approximation
Proceedings of the Conference held at
the Mathematical Research Institute
at Oberwolfach, Black Forest,
August 9 -16, 1980
Edited by
P.L. Butzer, Aachen
B. Sz.-Nagy, Szeged
E. Gorlich, Aachen
1981 Birkhauser Verlag
Basel· Boston· Stuttgart
Editors
Prof. Dr. P. L. Butzer
Rhein.-Westf. Techn. Hochschule Aachen
Lehrstuhl A flir Mathematik
Templergraben 55
D-51 Aachen (FRG)
Prof. Dr. B. Sz.-Nagy
J6zsef Altila Tudom{myegyetem
Aradi vertanuk tere I
H-6720 Szeged (Hungary)
Prof. Dr. E. G6rlich
Rhein.-Westf. Techn. Hochschule Aachen
Lehrstuhl A flir Mathematik
Templergraben 55
D-51 Aachen (FRG)
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Functional analysis and approximation : proceedings
of the conference held at the Math. Research Inst.
at Oberwolfach, Black Forest, August 9-16,
1980 / ed. by P. L. Butzer ... - Basel ; Boston;
Stuttgart: Birkhauser, 1981.
(International series of numerical
mathematics; Vol. 60)
ISBN-13: 978-3-0348-9371-8 e-ISBN-13: 978-3-0348-9369-5
001: 10.1007/978-3-0348-9369-5
NE: Butzer, Paul L. [Hrsg.]; Mathematisches
Forschungsinstitut <Oberwolfach); GT
All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or
transmitted in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of the copyright owner.
© 1981 Birkhauser Verlag Basel
Softcover reprint of the hardcover I st edition 198 I
In Memory of
Jacob Lionel Bakst Cooper
Born on December 27, 1915 in Beaufort-West, South Africa
Died on August 8, 1979 in London
7
Preface
These Proceedings form a record of the lectures presented at the interna
tional Conference on Functional Analysis and Approximation held at the Ober
wolfach Mathematical Research Institute, August 9-16, 1980. They include 33 of
the 38 invited conference papers, as well as three papers subsequently submitted
in writing. Further, there is a report devoted to new and unsolved problems,
based on two special sessions of the conference. The present volume is the sixth
Oberwolfach Conference in Birkhauser's ISNM series to be edited at Aachen *. It
is once again devoted to more significant results obtained in the wide areas of
approximation theory, harmonic analysis, functional analysis, and operator
theory during the past three years. Many of the papers solicited not only outline
fundamental advances in their fields but also focus on interconnections between
the various research areas.
The papers in the present volume have been grouped into nine chapters.
Chapter I, on operator theory, deals with maps on positive semidefinite opera
tors, spectral bounds of semigroup operators, evolution equations of diffusion
type, the spectral theory of propagators, and generalized inverses. Chapter II, on
functional analysis, contains papers on modular approximation, interpolation
spaces, and unconditional bases. In Chapter III, on abstract harmonic analysis,
one may find results on approximation on compact abelian groups, minimal
projections in LI, Wiener type distributions, and analysis on local fields, whereas
Chapter IV, on Fourier analysis and integral transforms, comprises papers on
polynomial inequalities, classical orthogonal expansions, multiple series, and the
Hilbert transform. Chapter V deals with best approximation, in general Hilbert
spaces, in the complex domain, as well as in the multipoint sense. Chapter VI, on
approximation by linear operators, includes an estimate for the Lebesgue
function of Lagrange interpolation, a uniform boundedness theorem with rates,
slow and asymptotically optimal approximations. Strong and Muntz approxima
tion then follow in Chapter VII, whereas problems of asymptotic distribution of
lattice points as well as two papers concerned with limit theorems of probabilty
theory in Banach spaces appear in Chapter VIII. Chapter IX contains papers on
spline functions and piecewise polynomial approximation as well as a paper on
dominant integrability. The volume closes with a bibliography on Bernstein
polynomials, as well as the section on 22 new and unsolved problems.
One mathematician was sorely missed at the conference. Lionel Cooper,
who had actively taken part in all but one of our conferences since 1963, was
again on the list of distinguished speakers who were invited. But in August of
1979 he passed away after heart operation. The loss caused by his death will
surely be long felt by the scientific world, in particular by the community of
8 Preface
mathematicians and physicists. The participants and organizing committee of the
Conference wish to dedicate these Proceedings to the memory of this distin
guished and independent scientist. Lionel Cooper also was a sincere friend to
many of us. Two brief appreciations of his life and work appear in these Proceed
ings.
The editors' warm thanks are due to all of the participants and contribu
tors: they made the conference the success it was; to Wolfgang Splettstosser for
his competent handling of the greater part of the general editorial work; to Rolf
1. Nessel for valuable advice during the preparations of the conference; to the
coworkers and research assistants from Aachen for their help in organizing the
conference, and to the secretaries of Lehrstuhl A fUr Mathematik for retyping
many of the papers and for their aid in preparing this volume. To Carl Einsele of
Birkhauser Verlag, Basel, we extend our thanks for his cooperation over the
years.
Aprill98l
P.L. Butzer E. Gorlich B.Sz.-Nagy
Aachen Aachen Szeged
* The earlier volumes are:
1. On Approximation Theory. Oberwolfach 1963. Eds.: P.L. Butzer and J. Korevaar. ISNM,
vol. 5, Basel 1964 (second edition 1972), XVI + 261 pages.
2. Abstract Spaces and Approximation. Oberwolfach 1969. Eds.: P.L. Butzer and B.Sz.-Nagy.
ISNM, vol. 10, Basel 1969,423 pages.
3. Linear Operators and Approximation I. Oberwolfach 1971. Eds.: P.L. Butzer, J.P. Kahane
and B. Sz.-Nagy. ISNM, vol. 20, Basel 1972,506 pages.
4. Linear Operators and Approximation II. Oberwolfach 1974. Eds.: P.L. Butzer and B.Sz.
Nagy. ISNM, vol. 25, Basel 1974,585 pages.
5. Linear Spaces and Approximation. Oberwolfach 1977. Eds.: P. L. Butzer and B. Sz.-Nagy.
ISNM, vol. 40, Basel 1978,685 pages.
9
Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Zur Tagung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
List of participants ............................................... 12
Program of the Sessions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
P. L. Butzer: Jacob Lionel Bakst Cooper - in memoriam ................ 19
A.J. W. Hill: A testimony from a friend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
I Operator Theory
T. Ando: Fixed points of certain maps on positive semidefinite operators 29
M. Wolff: A remark on the spectral bound of the generator of semigroups
of positive operators with applications to stability theory. . . . . . . . . . . 39
G. Lumer: Local operators, regular sets, and evolution equations of
diffusion type .............................................. 51
P. Masani: An outline of the spectral theory of propagators ............. 73
M.Z. Nashed: On generalized inverses and operator ranges ..... . . . . . . . . 85
II Functional Analysis
J. Musielak: Modular approximation by a filtered family of linear opera-
~ ....................................................... ~
C. Bennett, R. Sharpley: Interpolation between HI and Lao . . . . . . . . . . . . .. 111
C. Ciesielski: The Franklin orthogonal system as unconditional basis in
ReHI and VMO ............................................ 117
III Abstract Harmonic Analysis
H. Ombe, C. W. Onneweer: Bessel potential spaces and generalized Lip-
schitz spaces on local fields ................................... 129
P. Lambert: On the minimum norm property of the Fourier projection in
Ll-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 139
H. G. Feichtinger: Banach spaces of distributions of Wiener's type and
interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 153
W.R. Bloom: Approximation theory on the compact solenoid ........... 167
IV Fourier Analysis and Integral Transforms
J. Szabados: Bernstein and Markov type estimates for the derivative of a
polynomial with real zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 177
E. Gorlich, C. Markett: Projections with norms smaller than those of the
ultraspherical and Laguerre partial sums. . . . . . . . . . . . . . . . . . . . . . .. 189
F. M6ricz: The regular convergence of multiple series . . . . . . . . . . . . . . . . .. 203
10 Contents
B. Muckenhoupt: Norm inequalities relating the Hilbert transform to the
Hardy-Littlewood maximalfunction ........................... 219
M. Wehrens: Best approximation on the unit sphere in Rk ............. " 233
V Best Approximation
H. Berens: Ein Problem uber die beste Approximation in Hilbertraumen .. 247
U. Westphal: Ober Existenz- und Eindeutigkeitsmengen bei der besten
Ko-Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 255
L. Bijvoets, W. Hogeveen, J. Korevaar: Inverse approximation theorems of
Lebedev and Tamrazov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 265
R. Beatson, Ch. K. Chui: Best multipoint local approximation ........... 283
VI Approximation by Linear Operators
P. Erdos, P. Vertesi: On the Lebesgue function of interpolation .......... 299
W. Dickmeis, R.J. Nessel: A uniform boundedness principle with rates and
an application to linear processes .............................. 311
P. C. Sikkema: Slow approximation with convolution operators . . . . . . . . .. 323
E. <;i"oriich: A remark on asymptotically optimal approximation by Faber
series ..................................................... 335
VII Strong and Miintz Approximation
L. Leindler: Strong approximation and generalized Lipschitz classes. . . . .. 343
V. Totik: Strong approximation and the behaviour of Fourier series ...... 351
D. Leviatan: On the rate of approximation by Muntz polynomials satisfy-
ing constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 365
VIII Number Theory and Probability
P. D. Lax, R. S. Phillips: The asymptotic distribution of lattice points in
Euclidean and non-Euclidean spaces. . . . . . . . . . . . . . . . . . . . . . . . . .. 373
V. Paulauskas: On the approximation of indicator functions by smooth
functions in Banach spaces ................................... 385
M. Roeckerath: On the o-closeness of the distribution of two weighted
sums of Banach space valued martingales with applications ........ 395
IX Splines and Numerical Integration
W. Schempp: Approximation und Transformationsmethoden III ........ 409
J.B. Kioustelidis: Uniqueness of optimal piecewise polynomial Ll approx-
imations for generalized convex functions . . . . . . . . . . . . . . . . . . . . . .. 421
N. S. Murthy, C. F. Osgood, O. Shisha: The dominated integral of functions
of two variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 433
E.L. Stark: Bemstein-Polynome, 1912-1955 .......................... 443
New and unsolved problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 463
Errata ........................................................ " 474
Alphabetical list of papers ....................................... " 476
Mathematics subject classification numbers. . . . . . . . . . . . . . . . . . . . . . . . . .. 478
Key words and phrases ........................................... 481
II
ZurTagung
Yom 9. bis l6.August 1980 fand im Mathematischen Forschungsinstitut Ober
wolfach eine Tagung tiber «Funktionalanalysis und Approximation» statt. Sie
setzte die 1963 begonnene und inzwischen zur Tradition gewordene Reihe
internationaler Tagungen tiber Approximationstheorie und angrenzende Gebiete
fort. Djesmal stand sie unter der Leitung von Prof. P. L. Butzer (Aachen), Prof.
E. Gorlich (Aachen) und Prof. B. Szokefalvi-Nagy (Szeged, Ungarn). Es nahmen
54 Mathematiker aus 14 Nationen an der Tagung teil, darunter auch viele
Kollegen, die zum ersten Mal eine Konferenz dieser Reihe besuchten, insbeson
dere mehrere jtingere Mathematiker. Zum Bedauern aller Teilnehmer muBten
eine Reihe von Kollegen aus der UdSSR ihre Zusage in letzter Minute zurtickzie
hen.
Das Vortragsprogramm bestand aus 38 Ubersichts- und Spezialvortragen, in
denen ein breites Spektrum von Themen aus den verschiedensten Gebieten der
Approximationstheorie, der harmonischen Analysis, der Funktionalanalysis und
der Operatortheorie behandelt wurden. Zwei weitere Sitzungen waren aktuellen
Problemstellungen gewidmet; hier wurden von den Teilnehmern 18 neue und
ungeloste Probleme vorgestellt. (Der Programmablauf ist auf den Seiten 15-17
ausfUhrlich wiedergegeben.) Der vorliegende Band enthalt den groBten Teil
dieser Vortrage u~d Problemstellungen.
Neben dem Vortragsprogramm fanden zwei gesellige Abende statt, und am
Mittwochnachmittag das traditionelle Ausflugsprogramm nach Baden-Baden,
Freiburg, Freudenstadt und in die nahere Umgebung.
Die Tagung war gekennzeichnet durch eine kollegiale und freundschaftliche
Atmosphare, wozu die Teilnehmer durch ihr spontanes und sympatisches
Mitwirken in vielfaltiger Weise beigetragen haben. Allen Vortragenden, den
Sitzungsleitern und besonders den Vorsitzenden der beiden «problem sessions»
sei fUr ihr Engagement herzlich gedankt.
An dieser Stelle ist besonders die Gastfreundschaft und Hilfsbereitschaft der
Mitarbeiter des Oberwolfacher Instituts zu erwahnen, ohne die soleh eine
Tagung kaum denkbar ware, und fUr die sich die Tagungsleiter bei den Damen
und Herren des Oberwolfacher Hauses und insbesondere bei dem Direktor des
Instituts, Herrn Professor Dr. M. Barner, herzlich bedanken mochten.
Tagungsleiter: P.L. Butzer E. Gorlich B.Sz.-Nagy