Approximation of multiple integrals by simple integrals involving periodic functions

In this paper, we consider problems of numerical integration of fast oscillatory functions of one variable, obtained by using α‐dense curves and approximating multiple integrals. Using first, periodic and regular α‐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.