Semi-Groups of Operators and Approximation

331 Pages·1967·11 MB·English

by  Professor Dr. Paul P. Butzer

Die Grundlehren der
mathematischen Wissenschaften
in Einze1darstellungen
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 145
Herausgegeben von
J.L.Doob . E.Heinz· F.Hirzebruch· E.Hopf. H.Hopf
W.Maak· S. Mac Lane . W.Magnus· D.Mumford
M. M. Postnikov . F. K. Schmidt· D. S. Scott· K. Stein
C;esch~t~uorende Herausgeber
B. Eckmann und B. L. van der Waerden
Paul L.Butzer . Hubert Berens
Semi-Groups of Operators
and Approximation
Springer-Verlag New York Inc. 1967
Professor Dr. Paul L. Butzer
Rheinisch·Westfa.lische Technische Hochschule
Aachen
Dr. rer. nat. Hubert Berens
Rheinisch-\\lestfalische Technische Hochschule
Aachen
Geschaftsfiihrende Herausgeber:
Professor Dr. B. Eckmann
Eidgenossische Technische Hochschule Zurich
Professor Dr. B. L. van der Waerden
Mathematisches Institut der Universitat Zurich
ISBN-13: 978-3-642-46068-5 e-ISBN-I3: 978-3-642-46066-1
DOl: 10.1007/978-3-642-46066-1
All righ ts reserved
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag
© by Springer-Verlag, Berlin· Heidelberg 1967
Softcover reprint of the hardcover 1st edition 1967
Library of Congress Catalog Card Number 68-II980
Title NO.5 128
Preface
In recent years important progress has been made in the study of
semi-groups of operators from the viewpoint of approximation theory.
These advances have primarily been achieved by introducing the
theory of intermediate spaces. The applications of the theory not only
permit integration of a series of diverse questions from many domains
of mathematical analysis but also lead to significant new results on
classical approximation theory, on the initial and boundary behavior
of solutions of partial differential equations, and on the theory of singular
integrals.
The aim of this book is to present a systematic treatment of semi
groups of bounded linear operators on Banach spaces and their connec
tions with approximation theoretical questions in a more classical
setting as well as within the setting of the theory of intermediate spaces.
However, no attempt is made to present an exhaustive account of the
theory of semi-groups of operators per se, which is the central theme
of the monumental treatise by HILLE and PHILLIPS (1957). Neither
has it been attempted to give an account of the theory of approximation
as such. A number of excellent books on various aspects of the latter
theory has appeared in recent years, so for example CHENEY (1966),
DAVIS (1963), LORENTZ (1966), MEINARDUS (1964), RICE (1964), SARD
(1963). By contrast, the present book is primarily concerned with those
aspects of semi-group theory that are connected in some way or other
with approximation. Special emphasis is placed upon the significance
of the relationships between the abstract theory and its various applica
tions. This is, in fact, the original aim of the Springer Grundlehren as
suggested by the subtitle of the series.
The present book is written for the graduate student as well as for
the research mathematician. It can be read by one who is familiar
with real variable theory and the elements of functional analysis. To
make the exposition self-contained these foundations are collected in
the Appendix. The results given are not always presented in their
most general form, so that the reader is not distracted by many of the
possible, but often irrelevant, complications. Furthermore, an attempt
has been made to make the presentation and proofs of the theorems
as clear and detailed as practicable so that the book will, in fact, be
accessible to the student reader. About two-thirds of the material is
treated here for the first time outside of technical papers, and about a
half is based upon recent research. Each chapter concludes with a
detailed section entitled "Notes and Remarks", containing references
vi Preface
and appropriate historical remarks to the principal results treated,
as well as information on important topics related to, but not included
among those given in the body of the text. In this way the book may
furnish additional information for the research mathematician. Any
inaccuracy or omission in assigning priorities for important discoveries
is unintentional and the writers will appreciate any corrections suggested
by colleagues in the field.
Chapter I gives the standard theory of semi-groups of operators
on Banach spaces, the presentation being straight-forward, systematic
and without unnecessary details, yet sufficiently complete to include
the major results. It aims to serve as an introduction to the theory.
Chapter II presents the basic approximation theorems for semi
group operators. Both direct and converse, optimal and non-optimal
approximation theorems for such operators are studied in Banach
spaces. Special emphasis is placed on the various concepts of generalized
derivatives as well as on the applications of the theory to the initial
and boundary behavior of solutions of partial differential equations,
in particular to the study of Dirichlet's problem for the unit disk and
Fourier's problem of the ring. The material of this chapter is largely
based upon the research efforts of the writers, initiated by the senior
author in 1956-1957, and continued jointly and individually since
1962.
Chapter III is devoted to the incorporation and grouping of the
powerful approximation theorems for semi-group operators, as discussed
in Chapter II, into the theory of intermediate spaces between the
initial Banach space and the domain of definition of powers of the
infinitesimal generator of the semi-group, and to deep generalizations
of the corresponding theorems in the new setting. These goals are
primarily achieved by the development of an interpolation method
between Banach spaces, in particular by the K- and J-methods of
JAAK PEETRE. There also are applications to Lorentz spaces and inter
polation theorems. Although the emphasis here is mainly on the role of
intermediate spaces in the study of semi-groups of operators, the theory
of intermediate spaces per se is developed to a limited extent. Conse
quently, Chapter III may provide an introduction to interpolation
space theory, especially since this theory, founded in 1959-1963, has
not been treated in book-form previously.
Chapter IV outlines and discusses some of the many applications
of the general theory presented in Chapter III, emphasizing the semi
group of left translations as well as the singular integrals of Abel
Poisson for periodic functions and of Cauchy-Poisson for functions
defined on the real line. Finally, the singular integral of Gauss-Weier
strass on Euclidean n-space is treated in connection with Sobolev and
Preface Vll
Besov spaces. This chapter stresses the interplay between functional
analysis and "hard" classical analysis such as the theory of Fourier
series and that of Fourier and Hilbert transforms.
The references cited are listed in the bibliography and a conventional
terminology is used so that it will not be necessary to continually refer
to a collection of symbols.
About ten years ago the late JEAN FAVARD suggested to the senior
author to attempt a book on approximation, based on the semi-group
approach. We are particularly grateful that he was able to participate
in the Conference on Approximation Theory conducted by one of us
at the Mathematical Research Institute, Oberwolfach, in August of
1963, before his untimely death in January, 1965. His sincere and abiding
interest and encouragement over the years are deeply acknowledged.
The senior author also wishes to express his gratitude to Professor
EINAR HILLE for the opportunity to participate in a highly profitable
seminar on semi-groups of operators conducted by Hille at the University
of Mainz in 1956/57. Professor JAAK PEETRE kindly sent us his many
papers on intermediate spaces even in their preprint form.
We are grateful to Miss URSULA WESTPHAL and Mr. KARL SCHERER
for their critical reading of the manuscript, assistance in reading the
proofs, and preparation of the index. We also wish to express our
appreciation to Mrs. DORIS EWERS for her patient and careful typing
of the manuscript. Last but not least, our warm thanks go to Professor
BELA SZ.-NAGY for his invitation to write this book for the Springer
Series as well as to the Springer-Verlag for producing it in accord with
their usual high standards of pUblication.
Aachen, June 1967
P. L. BUTZER' H. BERENS
Contents
Chapter One
Fundamentals of Semi-Group Theory
1.0 Introduction .
1.1 Elements of Semi-Group Theory 7
1.1.1 Basic Properties 7
1.1.2 Holomorphic Semi-Groups 15
1.2 Representation Theorems for Semi-Groups of Operators 18
1.2.1 First Exponential Formula 18
1.2.2 General Convergence Theorems. . . 24
1.2.3 'Weierstrass Approximation Theorem 28
1.3 Resolvent and Characterization of the Generator 30
1.3.1 Resolvent and Spectrum .. 30
1.3.2 Hille-Y osida Theorem . 34
1.3.3 Translations; Groups of Operators 38
1.4 Dual Semi-Groups 45
1.4.1 Theory ... 45
1.4.2 Applications 52
1.5 Trigonometric Semi-Groups 55
1.5.1 Classical Results on Fourier Series 56
1.5.2 Fourier's Problem of the Ring . 59
1.5.3 Semi-Groups of Factor Sequence Type 64
1.5.4 Dirichlet's Problem for the Unit Disk 69
1.6 Notes and Remarks 73
Chapter Two
Approximation Theorems for Semi-Groups of Operators
2.0 Introduction . 83
2.1 Favard Classes and the Fundamental Approximation Theorems 86
2.1.1 Theory. . . . . . . . . . . . . . . . . . . . 86
2.1.2 Applications to Theorems of Titchmarsh and Hardy-Littlewood. 92
2.2 Taylor, Peano, and Riemann Operators Generated by Semi-Groups of
Operators . . . . . . . . . .. .......... . 95
2.2.1 Generalizations of Powers of the Infinitesimal Generator 95
2.2.2 Saturation Theorems ... 102
2.2.3 Generalized Derivatives of Scalar-valued Functions. . . 106
Butzer/Berens, Semi .. Groups a
x Contents
2.3 Theorems of Non-optimal Approximation. 111
2.3.1 Equivalence Theorems for Holomorphic Semi-Groups 111
2.3.2 Lipschitz Classes . . . . . . . . . 116
2.4 Applications to Periodic Singular Integrals 117
2.4.1 The Boundary Behavior of the Solution of Dirichlet's Problem;
Saturation . . . . . . . . . . . . . . . . . . . . . . 117
2.4.2 The Boundary Behavior for Dirichlet's Problem; Non-optimal
Approximation .................... , 122
2.4.3 Initial Behavior of the Solution of Fourier's Ring Problem 127
2.5 Approximation Theorems for Resolvent Operators 130
2.5.1 The Basic Theorems. . . . . . . . . . . 130
2.5.2 Resolvents as Approximation Processes. . 136
2.6 Laplace Transforms in Connection with a Generalized Heat Equation 140
2.7 Notes and Remarks 147
Chapter Three
Intermediate Spaces and Semi-Groups
3.0 Scope of the Chapter. . . . . . . . . . . . . . . . 157
3.1 Banach Subspaces of X Generated by Semi-Groups of Operators 159
3.2 Intermediate Spaces and Interpolation . . . . . . . . . . . . 165
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 165
3.2.2 The K- and J-Methods for Generating Intermediate Spaces 166
3.2.3 On the Equivalence of the K- and J -Methods. 171
3.2.4 A Theorem of Reiteration . . . . 175
3.2.5 Interpolation Theorems . . . . . 179
3.3 Lorentz Spaces and Convexity Theorems 181
3.3.1 Lorentz Spaces ........ . 181
3.3.2 The Theorems of M. Riesz-Thorin and Marcinkiewicz 187
3.4 Intermediate Spaces of X and 0 (Ar) • • • . . . . • . • • 191
3.4.1 An Equivalence Theorem for the Intermediate Spaces X",ra . 191
3.4.2 Theorems of Reduction for the Spaces X",ra' ... , . .. 195
3.4.3 The Spaces X~,r:oo .•.•.. , ...•.....•. , 204
3.5 Equivalent Characterizations of XIJ<,r;q Generated by Holomorphic Semi-
Groups . . . . . . 207
3.6 Notes and Remarks 211
Chapter Four
Applications to Singular Integrals
4.0 Orientation . . 226
4.1 Periodic Functions 227
4.1.1 Generalized Lipschitz Spaces 227
4.1.2 The Singular Integral of Abel-Poisson 232
4.1.3 The Singular Integral of Weierstrass . 235
Contents xi
4.2 The Hilbert Transform and the Cauchy-Poisson Singular Integral 236
4.2.1 Foundations on the Fourier Transform • 236
4.2.2 The Hilbert Transform . . . . . . . . 240
4.2.3 The Singular Integral of Cauchy-Poisson 248
4.3 The Weierstrass Integral on Euclidean n-Space 254
4.3.1 Sobolev and Besov Spaces ... 254
4.3.2 The Gauss-Weierstrass Integral. 261
4.4 Notes and Remarks . . . . . . . . 270
Appendix 279
Bibliography. 295
Index .... 314