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Approximate wavelets and the approximation of pseudodifferential operators. (English) Zbl 0932.65149
In this interesting paper, the authors introduce an approximate multiresolution analysis for spaces generated by smooth functions providing high-order semi-analytic cubature formulas for multidimensional integral operators of mathematical physics. Since these functions satisfy refinement equations with any prescribed accuracy, wavelet-type methods can be applied. An approximate decomposition of the finest scale space into almost orthogonal wavelet spaces is obtained. For the Gaussian function, properties of analytic prewavelets are studied and the projections onto the wavelet spaces are described. The wavelets retain the property of the scaling function to provide efficient analytic expressions of important integral operators, which leads to sparse and semi-analytic representations of these operators.
Reviewer: M.Tasche (Rostock)

MSC:
65T60 Numerical methods for wavelets
35S05 Pseudodifferential operators as generalizations of partial differential operators
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
45P05 Integral operators
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
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