Stability of spline approximation methods for multidimensional pseudodifferential operators. (English) Zbl 0804.65141
The authors present a unique Banach algebra approach to the stability analysis of spline approximation methods for singular integral operators on the Euclidean space \(\mathbb{R}^ n\), which occur as local representatives of pseudodifferential operators on closed manifolds or on manifolds with a smooth boundary. Thus the paper may be regarded as a first step in establishing spline approximation methods for pseudodifferential equations on manifolds. After giving a treatise of the Fredholm theory in \(L_ p\) spaces, the authors study algebras of approximation sequences for large classes of convolution operators, leading to necessary and sufficient conditions for the stability of Galerkin, collocation and qualocation methods based on spline functions.
Reviewer: J.Elschner (Berlin)
MSC:
65R20 | Numerical methods for integral equations |
65J10 | Numerical solutions to equations with linear operators |
35S05 | Pseudodifferential operators as generalizations of partial differential operators |
47G10 | Integral operators |
47G30 | Pseudodifferential operators |
45P05 | Integral operators |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
Galerkin method; collocation method; Banach algebra; stability; spline approximation methods; singular integral operators; pseudodifferential operators; closed manifolds; Fredholm theory; convolution operators; qualocation methodsReferences:
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