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Trigonometric approximation of solutions of periodic pseudodifferential equations. (English) Zbl 0693.65093
Oper. Theory, Adv. Appl. 41, 359-383 (1989).
The Ritz-Galerkin reduction method with trigonometric polynomials as trial and test functions is used to construct approximate solutions of periodic singular integral equations and pseudodifferential equations. The analysis is extended to Hölder-Zygmund spaces which allow via imbedding sharp pointwise error estimates. Chapters 1-4 deal with the scalar case and can be read without further background concerning the generalizations in chapt. 5-6.
It is shown how to extend the analysis to a scalar pseudodifferential equation and to systems by using Bessel potentials. The necessary matrix factorization results are quoted.
The authors present results on pointwise rates of convergence showing among other things that if the solution is in \(C^ r\), then the error is \(O(n^{-r}\log n)\), where n is the degree of the trigonometric polynomials. This result is the best that can be expected since the same rate of convergence holds for the partial sums of the Fourier series of a function in \(C^ r\).

MSC:
65R20 Numerical methods for integral equations
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
42A10 Trigonometric approximation
45E05 Integral equations with kernels of Cauchy type