Spectral Theory of Approximation Methods for Convolution Equations
The aim of the present book is to propose a new algebraic approach to the study of norm stability of operator sequences which arise, for example, via discretization of singular integral equations on composed curves. A wide variety of discretization methods, including quadrature rules and spline or w...
Main Authors: | , , |
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Format: | eBook |
Language: | English |
Published: |
Basel
Birkhäuser
1995, 1995
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Edition: | 1st ed. 1995 |
Series: | Operator Theory: Advances and Applications
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Subjects: | |
Online Access: | |
Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 1 Invertibility in Banach algebras
- 1.1 Banach algebras and C*-algebras
- 1.2 Linear operators
- 1.3 Stability of operator sequences
- 1.4 Local principles
- 1.5 The finite section method for Toeplitz operators
- 1.6 A general invertibility scheme
- 1.7 Norm-preserving localization
- 1.8 Exercises
- 1.9 Comments and references
- 2 Spline spaces and Toeplitz operators
- 2.1 Singular integral operators-constant coefficients
- 2.2 Piecewise constant splines
- 2.3 Algebras of Toeplitz operators (Basic facts)
- 2.4 Discretized Mellin convolutions
- 2.5 Algebras of Toeplitz operators (Fredholmness)
- 2.6 General spline spaces
- 2.7 Spline projections
- 2.8 Canonical prebases
- 2.9 Concrete spline spaces
- 2.10 Concrete spline projections
- 2.11 Approximation of singular integral operators
- 2.12 Proofs
- 2.13 Exercises
- 2.14 Comments and references
- 3 Algebras of approximation sequences
- 3.1 Algebras of singular integral operators
- 6.3 Around spline approximation methods
- 3.2 Approximation using piecewise constant splines
- 3.3 Approximation of homogeneous operators
- 3.4 The stability theorem
- 3.5 Basic properties of approximation sequences
- 3.6 Proof of the stability theorem
- 3.7 Sequences of local type
- 3.8 Concrete approximation methods
- 3.9 Exercises
- 3.10 Comments and references
- 4 Singularities
- 4.1 Approximation of operators in Toeplitz algebras
- 4.2 Multiindiced approximation methods
- 4.3 Approximation of singular integral operators
- 4.4 Approximation of compound Mellin operators
- 4.5 Approximation over unbounded domains
- 4.6 Exercises
- 4.7 Comments and references
- 5 Manifolds
- 5.1 Algebras of singular integral operators
- 5.2 Splines over homogeneous curves
- 5.3 Splines over composed curves
- 5.4 The stability theorem
- 5.5 A Galerkin method
- 5.6 Exercises
- 5.7 Comments and references
- 6 Finite sections
- 6.1 Finite sections of singular integrals
- 6.2 Finite sections of discrete convolutions