Optimal approximation formulas for computing of Fourier integrals in a Hilbert space

Аннотация

In most cases it is impossible to get the exact values of such integrals. Thus, they can be approximately calculated using the formulas of numerical integration. However, standard methods of numerical integration cannot be successfully applied for that. Therefore special effective methods should be developed for approximation of highly oscillating integrals. One of the first numerical integration formula for the integral (1) was obtained by Filon [5] in 1928 using a quadratic spline. Since then, for integrals of different types of highly oscillating functions many special effective methods have been developed, such as Filon-type method, Clenshaw-Curtis- Filon type method, Levin type methods, modified Clenshaw-Curtis method, generalized quadrature rule, and Gauss-Laguerre quadrature (see, for example [4,7,8] and references therein).