Accuracy-optimal evaluation of the fourier transform for C_L,N,ɛ functions

Abstract

We consider minimum-error evaluation of integrals of rapidly oscillating functions of the form

$$I(\omega ) = \int\limits_a^b {f(x)\left\{ {\begin{array}{*{20}c} {\sin \omega x} \\ {\cos \omega x} \\ \end{array} } \right\}dx}$$

where f(x) is in the class of interpolation Lipschitzian functions and the information given about f(x) is approximate. The boundary function method is applied to derive lower bounds on the numerical integration error of l(ω) in this class and quadrature formulas attaining these bounds are constructed.