A Fourier series approach to cubature formulas for integrals on the disk witha non-smooth weight function

We study a new cubature formula for the evaluation of an integral over the disk with respect to a weight function. The method is based on an analysis of the Fourier series of the weight function and a reduction of the bivariate integral into an infinite sum of univariate integrals. Several experimental results with non-smooth weight functions show that the accuracy of the method is superior to standard cubature formulas on the disk; this is natural to expect since the new formula is adapted to treat singularities of maximal order. Error estimates provide the theoretical basis for the good performance of the new algorithm.