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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...

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Bibliographic Details
Main Authors: Hagen, Roland, Roch, Steffen (Author), Silbermann, Bernd (Author)
Format: eBook
Language:English
Published: Basel Birkhäuser 1995, 1995
Edition:1st ed. 1995
Series:Operator Theory: Advances and Applications
Subjects:
Numerical Analysis
Mathematical Analysis
Analysis
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

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