Equivalence of Nyström’s method and Fourier methods for the numerical solution of Fredholm integral equations

Berrut, Jean-Paul; Trummer, Manfred R.

doi:10.1090/S0025-5718-1987-0878694-8

Equivalence of Nyström’s method and Fourier methods for the numerical solution of Fredholm integral equations
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by Jean-Paul Berrut and Manfred R. Trummer PDF

Math. Comp. 48 (1987), 617-623 Request permission

Abstract:

Nyström’s method with the trapezoidal rule, and the Fourier method, produce the same approximation to the solution of an integral equation at the collocation points for Nyström’s method. This equivalence allows the derivation of error estimates for Nyström’s method, and gives an intuitive explanation for its good performance in the periodic case. The equivalence holds for Fourier methods with arbitrary orthogonal basis functions. The quadrature rule for numerical integration must have the collocation points as abscissae, and must yield the exact entries of the Gramian matrix of the orthogonal basis.

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