Full Description
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 wavelet approximations, is covered and studied from a unique point of view. The approach takes advantage of the fruitful interplay between approximation theory, concrete operator theory, and local Banach algebra techniques. The book is addressed to a wide audience, in particular to mathematicians working in operator theory and Banach algebras as well as to applied mathematicians and engineers interested in theoretical foundations of various methods in general use, particularly splines and wavelets. The exposition contains numerous examples and exercises. Students will find a large number of suggestions for their own investigations.
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.- 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.- 6.3 Around spline approximation methods.
