Product Integration with Application to Differential Equations

288 Pages·1984·9.154 MB·English

by  John Day Dollard, Charles N. Friedman

Product Integration
with Applications to
Differential Equations

GIAN-CARLO ROTA, Editor
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
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Product Integration
with Application to Differential Equations,
1979
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ENCYCLOPEDIA
OF MATHEMATICS
and Its Applications
GIAN-CARLO ROTA, Editor
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, Massachusetts
Editorial Board
Janos D. Aczel, Waterloo J. F. C. Kingman, Oxford
Richard Askey, Madison Donald E. Knuth, Stanford
Michael F. Atiyah, Oxford Peter D. Lax, Courant
Edwin F. Beckenbach, U.C.L.A. Joshua Lederberg, Rockefeller
Lipman Bers, Columbia Andre Lichnerowicz, College de France
Arne Beurling, Inst, for Advanced Study M. J. LighthiU, Cambridge
Garrett Birkhoff, Harvard Chia-Chiao Lin, M.I.I.
Salomon Bochner, Rice Jacques-Louis Lions, College de France
Raoul Bott, Harvard Roger Lyndon, Ann Arbor
Felix E. Browder, Chicago Marvin Marcus, Santa Barbara
A. P. Calderon, Buenos Aires N. Metropolis, Los Alamos Scientific Lab.
Peter A. Carruthers, Los Alamos Jan Mycielski, Boulder
S. Chandrasekhar, Chicago Steven A. Orszag, M.I.T.
S. S. Chern, Berkeley Alexander Ostrowski, Basle
Hermann Chernoff, M.I.T. Roger Penrose, Oxford
Paul Cohen, Stanford Carlo Pucci, Florence
P. M. Cohn, Bedford College, London C. R. Rao, Indian Statistical Institute
H. S. Mac Donald Coxeter, Toronto Fred S. Roberts, Rutgers
Nelson Dunford, Sarasota, Florida Abdus Salam, Trieste
F. J. Dyson, Inst, for Advanced Study M. P. Schiitzenberger, Paris
Giovanni Gallavotti, Rome Jacob T. Schwartz, Courant
Andrew M. Gleason, Harvard Irving Segal, M.I.T.
A. Gonzalez Dominguez, Buenos Aires Olga Taussky, Caltech
M. Gordon, Essex Rene Thorn, Bures-sur-Yvette
Nathan Jacobson, Yale John Todd, Caltech
Mark Kac, Rockefeller John W. Tukey, Princeton
Shizuo Kakutani, Yale Stanislaw Ulam, Gainesville; Los Alamos
Robert Kalaba, U.S.C. Veeravalli S. Varadarajan, U.C.L.A.
Samuel Karlin, Stanford Antoni Zygmund, Chicago
GIAN-CARLO ROTA, Editor
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Volume 10
Section: Analysis
Felix E. Browder, Section Editor
Product Integration
with Applications to
Differential Equations
John D. DoUard and Charles N. Friedman
Department of Mathematics
University of Texas, Austin, Texas
Foreword by
Felix E. Browder
University of Chicago
Appendix by
P. R. Masani
University of Pittsburgh
•
••
1979
Addison-Wesley Publishing Company
Advanced Book Program
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First published 1979 by Addison Wesley
First published 1979 by Addison Wesley
First published by Cambridge University Press 1984
First published by Cambridge University Press 1984
A catalogue record for this publication is available from the British Library
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Library of Congress Cataloguing in Publication data
Dollard, John Day.
Dollard, John Day.
Product integration with applications to
diffPerroednuticatl ienqteugartaiotinosn. with applications to
differential equations.
(Encyclopedia of mathematics and its applica
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III.I . FSrieerdiems.a n, Charles N., joint author. II. Title.
QIIAI. 37S1.eDri6e4s. 515'.35 79-20454
QA371.D64 515'.35 79-20454
ISBN 978-0-521-30230-2 Hardback
ISBN 978-0-521-30230-2 Hardback
Cambridge University Press has no responsibility for the persistence or
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Contents
Editor's Statement . . . . . . . . . . . . . . xi
Section Editor's Foreword . . . . . . . . . xiii
Preface . . . . . . . . . . . . . . . . . xix
Chapter 1 Product Integration of Matrix-Valued Functions . . .1
0 Introduction . . . . . . . . . . . . . . 1
1.1 Product Integration . . . . . . . . . . . . . . 3
1.2 Product Integral Analysis of Linear Ordinary Differ
ential Equations . . . . . . . . . . . . . . . . 12
1.3 Further Properties of Product Integrals . . . . . . 19
1.4 Estimates of Size, and the Product Integral as a Time-
Ordered Exponential . . . . . . . . . . . . . 28
1.5 Dependence on a Parameter . . . . . . . . . 32
1.6 Improper Product Integration . . . . . . . . . . 37
1.7 Alternative Definitions of the Product Integral . . . 50
1.8 Lebesgue-Integrable Functions . . . . . . . . 54
Notes to Chapter 1 . . . . . . . . . . 56
Chapter 2 Contour Product Integration . . . . . . . . 59
0 Introduction . . . . . . . . . . . . . 59
2.1 The Definition of Contour Product Integrals . . . . 59
2.2 The Product Integral of an Analytic Function and the
Analogues of Cauchy's Integral Theorem . . . . 62
2.3 A Cauchy Integral Formula for Product Integrals . . 66
2.4 Generalizations . . . . . . . . . . . . 77
Notes to Chapter 2 . . . . . . . . . . . . . 77
Chapter 3 Strong Product Integration . . . . . . . . . . 79
0 Introduction . . . . . . . . . . . . . . 79
3.1 Direct Extensions of the Results of Chapter 1 . . .80
vii
viii Contents
3.2 Generalization . . . . . . . . . . . . 83
3.3 The Space Lj (a, b\ ®{X)) . . . . . . . . . . . 86
3.4 Solution of Integral Equations . . . . . . . . . 94
3.5 Product Integration of Functions in L\{a,b) . . . . 98
3.6 Product Integrals Involving Unbounded Operators .108
Notes to Chapter 3 . . . . . . . . . . . . 119
Chapter 4 Applications . . . . . . . . . . . . . 121
4.1 Asymptotics for the Schrodinger Equation . . . .121
4.2 Weyl's Limit-Circle Classification . . . . . . . . 132
4.3 The Lie Product Formula . . . . . . . . . 135
4.4 The Hille-Yosida Theorem . . . . . . . . . . . 138
4.5 An Example Involving Unbounded Operators with
Variable Domain . . . . . . . . . . . . 144
Notes to Chapter 4 . . . . . . . . . . . . . 148
Chapter 5 Product Integration of Measures . . . . . . . 151
5.1 Introduction . . . . . . . . . . . . . 151
5.2 The Product Integral . . . . . . . . . . . . . . 154
5.3 Integral and Differential Equations . . . . . . . . 159
5.4 Further Properties of Product Integrals . . . . . . . . . 171
5.5 Improper Product Integration . . . . . . . . . . . . . . . 175
5.6 The Schrodinger Equation . . . . . . . . . . 178
5.7 The Equation y" + p(dx)y' + q(dx)y = 0 . . . .182
Notes to Chapter 5 . . . . . . . . . . . . 184
Chapter 6 Complements; Other Work and Further Results on
Product Integration . . . . . . . . . . . . . 187
Appendix I Matrices . . . . . . . . . . . . . . . 193
A.I.I Elementary Definitions . . . . . . . . . 193
A.1.2 Calculus of C -Valued Functions . . . . . 195
nxw
A.1.3 The Canonical Form of a Matrix . . . . . . . . . 196
A.1.4 The Spectrum of a Matrix . . . . . . . . . . . . 198
A.1.5 Some Additional Results . . . . . . . . . . . . . 201
References . . . . . . . . . . . . . . . . 205
Notes to the References . . . . . . . . . . 205