Approximation of additive convolution-like operators: Real C-star-algebra approach

320 Pages·2008·3.02 MB·English

by  Didenko V.

Frontiers in Mathematics
Advisory Editorial Board
Leonid Bunimovich (Georgia Institute of Technology, Atlanta, USA)
Benoît Perthame (Ecole Normale Supérieure, Paris, France)
Laurent Saloff-Coste (Cornell University, Rhodes Hall, USA)
Igor Shparlinski (Macquarie University, New South Wales, Australia)
Wolfgang Sprössig (TU Bergakademie, Freiberg, Germany)
Cédric Villani (Ecole Normale Supérieure, Lyon, France)
Victor D. Didenko
Bernd Silbermann
Approximation
of
Additive
Convolution-Like
Operators
Real C*-Algebra Approach
Birkhäuser Verlag
Basel . Boston . Berlin
Authors:
Victor D. Didenko Bernd Silbermann
Department of Mathematics Fakultät für Mathematik
University of Brunei Darussalam TU Chemnitz
Gadong BE 1410 09107 Chemnitz
Brunei Germany
e-mail: [e-mail] e-mail: [e-mail]
[e-mail]
2000 Mathematical Subject Classification: 31-xx, 35-xx,45-xx, 46-xx, 47-xx, 65-xx,
74-xx, 76-xx, 76-xx. In particular: 31A10, 31A30, 35Q15, 35Q30, 45A05, 45B05, 45Exx,
45L05, 45P05, 46N20, 46N40, 47B35, 47Gxx, 47N40, 65E05, 65Jxx, 65R20, 74B10, 74G15,
74Kxx, 74S15, 76D05
Library of Congress Control Number: 2008925217
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-7643-8750-1 Birkhäuser Verlag AG, Basel · Boston · Berlin
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Contents
Preface ix
1 Complex and Real Algebras 1
1.1 Complex and Real C∗-Algebras . . . . . . . . . . . . . . . . . . . . 1
1.2 Real Extensions of Complex ∗-Algebras . . . . . . . . . . . . . . . 4
1.3 Uniqueness of Involution in Real Extensions of Complex ∗-Algebras 7
1.4 Real and Complex Spectrum. Inverse Closedness . . . . . . . . . . 15
1.5 Moore-PenroseInvertibility in Algebra A˜ . . . . . . . . . . . . . . 20
1.6 Operator Sequences: Stability . . . . . . . . . . . . . . . . . . . . . 24
1.7 Asymptotic Moore-PenroseInvertibility . . . . . . . . . . . . . . . 31
1.8 Approximation Methods in Para-Algebras . . . . . . . . . . . . . . 38
1.9 Local Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.9.1 Gohberg-Krupnik Local Principle . . . . . . . . . . . . . . . 41
1.9.2 Allan’s Local Principle . . . . . . . . . . . . . . . . . . . . 44
1.9.3 Local Principle for Para-algebras . . . . . . . . . . . . . . . 45
1.10 Singular Integral and Mellin Operators . . . . . . . . . . . . . . . 47
1.10.1 Singular Integ(cid:1)rals. . . . . . . . . . . . . . . . . . . . . . . . 47
1.10.2 The Algebra p(α) . . . . . . . . . . . . . . . . . . . . . 49
1.10.3 Integral Repr(cid:1)esentations . . . . . . . . . . . . . . . . . . . . 51
1.10.4 The Algebra p(α,β,ν) . . . . . . . . . . . . . . . . . . . 52
1.10.5 Singular Integral Operators with Piecewise Continuous
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.10.6 Singular Integral Operators with Conjugation . . . . . . . . 56
1.11 Comments and References . . . . . . . . . . . . . . . . . . . . . . . 57
2 Integral Operators. Smooth Curves 59
2.1 Polynomial Galerkin Method . . . . . . . . . . . . . . . . . . . . . 60
2.2 Polynomial Collocation Method . . . . . . . . . . . . . . . . . . . . 65
2.3 Equations with Differential Operators . . . . . . . . . . . . . . . . 69
2.3.1 Singular Integro-Differential Equations . . . . . . . . . . . . 69
2.3.2 Approximate Solution of Singular Integro-Differential
Equations with Conjugation . . . . . . . . . . . . . . . . . . 72
vi Contents
2.4 A C∗-Algebra of Operator Sequences . . . . . . . . . . . . . . . . 73
2.4.1 Moore-PenroseInvertibility . . . . . . . . . . . . . . . . . . 86
2.5 The ε-Collocation Method . . . . . . . . . . . . . . . . . . . . . . . 86
2.6 Spline Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . 91
2.7 Quadrature Methods on the Unit Circle . . . . . . . . . . . . . . . 93
2.8 Polynomial Qualocation Method . . . . . . . . . . . . . . . . . . . 100
2.9 Spline Qualocation Method . . . . . . . . . . . . . . . . . . . . . . 104
2.10 Quadrature Methods on Closed Smooth Curves . . . . . . . . . . . 113
2.11 A Remark on Tensor Product Techniques . . . . . . . . . . . . . . 117
2.12 Comments and References . . . . . . . . . . . . . . . . . . . . . . . 119
3 Riemann-Hilbert Problem 123
3.1 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.2 Interpolation Method. Continuous Coefficients . . . . . . . . . . . 131
3.3 Local Principle for the Para-AlgebraA/J . . . . . . . . . . . . . . 136
3.4 Discontinuous Coefficients . . . . . . . . . . . . . . . . . . . . . . . 147
3.5 Generalized Riemann-Hilbert-Poincar´eProblem . . . . . . . . . . . 152
3.6 Comments and References . . . . . . . . . . . . . . . . . . . . . . . 154
4 Piecewise Smooth and Open Contours 157
4.1 Stability of Approximation Methods . . . . . . . . . . . . . . . . . 157
4.1.1 Quadrature Methods for Singular Integral Equations with
Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.1.2 The ε-Collocation Method . . . . . . . . . . . . . . . . . . . 168
4.1.3 Qualocation Method . . . . . . . . . . . . . . . . . . . . . . 170
4.2 Fredholm Properties of Local Operators . . . . . . . . . . . . . . . 171
4.2.1 Quadrature Methods . . . . . . . . . . . . . . . . . . . . . . 172
4.2.2 The ε-Collocation Method . . . . . . . . . . . . . . . . . . . 176
4.2.3 Qualocation Method . . . . . . . . . . . . . . . . . . . . . . 179
4.3 L2-Spaces with Weight . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.3.1 A Quadrature Method and Its Stability . . . . . . . . . . . 180
4.3.2 Local Operators and Their Indices . . . . . . . . . . . . . . 184
4.3.3 The Vanishing of the Index of Local Operators . . . . . . . 196
4.3.4 Modification of the Quadrature Method by Cutting Off
Corner Singularities . . . . . . . . . . . . . . . . . . . . . . 199
4.4 Double Layer Potential Equation . . . . . . . . . . . . . . . . . . . 202
4.4.1 Quadrature Methods . . . . . . . . . . . . . . . . . . . . . . 203
4.4.2 The ε-Collocation Method . . . . . . . . . . . . . . . . . . . 206
4.4.3 Qualocation Method . . . . . . . . . . . . . . . . . . . . . . 207
4.5 Mellin Operators with Conjugation . . . . . . . . . . . . . . . . . . 208
4.5.1 A Quadrature Method . . . . . . . . . . . . . . . . . . . . . 210
4.5.2 Fredholm Properties of Local Operators . . . . . . . . . . . 213
4.5.3 Index of the Local Operators . . . . . . . . . . . . . . . . . 218
4.5.4 Examples of the Essential Spectrum of Local Operators . . 226
Contents vii
4.6 Comments and References . . . . . . . . . . . . . . . . . . . . . . . 230
5 Muskhelishvili Equation 233
5.1 Boundary Problems for the Biharmonic Equation . . . . . . . . . . 233
5.1.1 Reduction of Biharmonic Problemsto a BoundaryProblem
for Two Analytic Functions . . . . . . . . . . . . . . . . . . 234
5.1.2 Reduction of Boundary Problems for Two Analytic
Functions to Integral Equations . . . . . . . . . . . . . . . . 241
5.2 Fredholm Properties. Invertibility . . . . . . . . . . . . . . . . . . . 244
5.3 Approximations on Smooth Contours. . . . . . . . . . . . . . . . . 256
5.3.1 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . 257
5.3.2 The ε-Collocation Method . . . . . . . . . . . . . . . . . . . 259
5.3.3 Qualocation Method . . . . . . . . . . . . . . . . . . . . . . 261
5.3.4 Biharmonic Problem . . . . . . . . . . . . . . . . . . . . . . 262
5.4 Approximation Methods on Special Contours . . . . . . . . . . . . 265
5.5 Galerkin Method. Piecewise Smooth Contour . . . . . . . . . . . . 270
5.6 Comments and References . . . . . . . . . . . . . . . . . . . . . . . 272
6 Numerical Examples 275
6.1 Muskhelishvili Equation . . . . . . . . . . . . . . . . . . . . . . . . 275
6.2 Biharmonic Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Bibliography 285
Index 303

Preface
Various aspects of numerical analysis for equations arising in boundary integral
equation methods have been the subject of several books published in the last 15
years [95, 102, 183, 196, 198]. Prominent examples include various classes of one-
dimensional singular integral equations or equations related to single and double
layerpotentials.Usually, a mathematicallyrigorousfoundation anderroranalysis
for the approximate solution of such equations is by no means an easy task. One
reason is the fact that boundary integral operators generally are neither integral
operatorsoftheformidentitypluscompactoperatornoridentityplusanoperator
with a small norm. Consequently, existing standard theories for the numerical
analysis of Fredholm integral equations of the second kind are not applicable. In
the last 15 years it became clear that the Banach algebra technique is a powerful
tool to analyze the stability problem for relevant approximation methods [102,
103, 183, 189]. The starting point for this approach is the observation that the
stabilityproblemisaninvertibilityprobleminacertainBanachorC∗-algebra.As
a rule, this algebrais verycomplicated – and one has to find relevantsubalgebras
to use such tools as local principles and representation theory.
However,invariousapplicationsthereoftenarisecontinuousoperatorsacting
on complex Banach spaces that are not linear but only additive – i.e.,
A(x+y)=Ax+Ay
for all x,y from a given Banach space. It is easily seen that additive operators
are R-linear provided they are continuous1. As an example, let us mention the
one-dimensional singular integral operators with conjugation often arising in me-
chanics. It is known that the study of such operators can be reduced to C-linear
operators, but with matrix-valued coefficients. In passing note that this observa-
tion is one of a number of motivations to study singular integral operators with
matrix-valued coefficients.
The present book is devoted to numerical analysis for certain classes of ad-
ditive operators and related equations, including singular integral operators with
conjugation,theRiemann-Hilbertproblem,Mellinoperatorswithconjugationand
the famous Muskhelishvili equation. Until now, most relevant material is only
1Hereandsubsequently,RandCdenotethefieldsofrealandcomplexnumbers,respectively.