Spectral Theory of Approximation Methods for Convolution Equations

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.