A new cubature formula with weight functions on the disc, with error estimates

TL;DR

This paper presents a novel cubature formula for disk integrals with weight functions, leveraging Fourier analysis to reduce the problem and demonstrating superior accuracy through experiments and error estimates.

Contribution

It introduces a new cubature method based on Fourier series analysis that improves accuracy over existing formulas for disk integrals.

Findings

The new formula outperforms standard methods in accuracy.

Error estimates support the method's reliability.

Experimental results confirm the theoretical advantages.

Abstract

We introduce a new type of 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 show that the accuracy of the method is superior to standard cubature formula on the disk. Error estimates provide the theoretical basis for the good performance of the new algorithm.