Approximation of multiple integrals by simple integrals involving periodic functions

Abstract

In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using a-dense curves and approximating multiple integrals. Using first, periodic and regular a-dense curves we propose a trapezoidal formula for calculating the periodic integrand obtained. Then, we consider the simple integrals as integrals with weight. We propose a method to evaluate the moments of the weight function. This allows us to build a recurrent formula for the orthogonal polynomials family and to use a Gaussian rule to estimate the simple integral. Finally, we adapt the Filon's method, consisting in evaluating the Fourier coefficients of a function, to the oscillatory integrand obtained by using reducing transformations.