Approximation of Additive Convolution-Like Operators

, by Didenko, Victor D.; Silbermann, Bernd

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ISBN: 9783764387501 | 3764387505
Cover: Paperback
Copyright: 7/4/2008

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This book deals with numerical analysis for certain classes of additive operators and related equations, including singular integral operators with conjugation, the Riemann-Hilbert problem, Mellin operators with conjugation, double layer potential equation, and the Muskhelishvili equation.

Preface	p. ix
Complex and Real Algebras	p. 1
Complex and Real C*-Algebras	p. 1
Real Extensions of Complex *-Algebras	p. 4
Uniqueness of Involution in Real Extensions of Complex *-Algebras	p. 7
Real and Complex Spectrum. Inverse Closedness	p. 15
Moore-Penrose Invertibility in Algebra A	p. 20
Operator Sequences: Stability	p. 24
Asymptotic Moore-Penrose Invertibility	p. 31
Approximation Methods in Para-Algebras	p. 38
Local Principles	p. 41
Gohberg-Krupnik Local Principle	p. 41
Allan's Local Principle	p. 44
Local Principle for Para-algebras	p. 45
Singular Integral and Mellin Operators	p. 47
Singular Integrals	p. 47
The Algebra [Sigma superscript p] ([alpha])	p. 49
Integral Representations	p. 51
The Algebra [Sigma superscript p] ([alpha],[beta],[nu])	p. 52
Singular Integral Operators with Piecewise Continuous Coefficients	p. 53
Singular Integral Operators with Conjugation	p. 56
Comments and References	p. 57
Integral Operators. Smooth Curves	p. 59
Polynomial Galerkin Method	p. 60
Polynomial Collocation Method	p. 65
Equations with Differential Operators	p. 69
Singular Integro-Differential Equations	p. 69
Approximate Solution of Singular Integro-Differential Equations with Conjugation	p. 72
A C*-Algebra of Operator Sequences	p. 73
Moore-Penrose Invertibility	p. 86
The [epilon]-Collocation Method	p. 86
Spline Galerkin Method	p. 91
Quadrature Methods on the Unit Circle	p. 93
Polynomial Qualocation Method	p. 100
Spline Qualocation Method	p. 104
Quadrature Methods on Closed Smooth Curves	p. 113
A Remark on Tensor Product Techniques	p. 117
Comments and References	p. 119
Riemann-Hilbert Problem	p. 123
Galerkin Method	p. 123
Interpolation Method. Continuous Coefficients	p. 131
Local Principle for the Para-Algebra A/J	p. 136
Discontinuous Coefficients	p. 147
Generalized Riemann-Hilbert-Poincare Problem	p. 152
Comments and References	p. 154
Piecewise Smooth and Open Contours	p. 157
Stability of Approximation Methods	p. 157
Quadrature Methods for Singular Integral Equations with Conjugation	p. 158
The [epilon]-Collocation Method	p. 168
Qualocation Method	p. 170
Fredholm Properties of Local Operators	p. 171
Quadrature Methods	p. 172
The [epilon]-Collocation Method	p. 176
Qualocation Method	p. 179
L[subscript 2]-Spaces with Weight	p. 180
A Quadrature Method and Its Stability	p. 180
Local Operators and Their Indices	p. 184
The Vanishing of the Index of Local Operators	p. 196
Modification of the Quadrature Method by Cutting Off Corner Singularities	p. 199
Double Layer Potential Equation	p. 202
Quadrature Methods	p. 203
The [epilon]-Collocation Method	p. 206
Qualocation Method	p. 207
Mellin Operators with Conjugation	p. 208
A Quadrature Method	p. 210
Fredholm Properties of Local Operators	p. 213
Index of the Local Operators	p. 218
Examples of the Essential Spectrum of Local Operators	p. 226
Comments and References	p. 230
Muskhelishvili Equation	p. 233
Boundary Problems for the Biharmonic Equation	p. 233
Reduction of Biharmonic Problems to a Boundary Problem for Two Analytic Functions	p. 234
Reduction of Boundary Problems for Two Analytic Functions to Integral Equations	p. 241
Fredholm Properties. Invertibility	p. 244
Approximations on Smooth Contours	p. 256
Galerkin Method	p. 257
The [epilon]-Collocation Method	p. 259
Qualocation Method	p. 261
Biharmonic Problem	p. 262
Approximation Methods on Special Contours	p. 265
Galerkin Method. Piecewise Smooth Contour	p. 270
Comments and References	p. 272
Numerical Examples	p. 275
Muskhelishvili Equation	p. 275
Biharmonic Problem	p. 281
Bibliography	p. 285
Index	p. 303
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Approximation of Additive Convolution-Like Operators

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