On the computation of the Fourier transform under the presence of nearby polar singularities (Q554273)

This paper is concerned with the computation of the Fourier transform \[ G(r)=\displaystyle{\int_{-\infty}^{\infty}g(t)e^{2irt}\,dt,} \] by keeping the difficulty of the presence of polar singularities near the real line but now dropping off the assumption that \(g\) vanishes outside of a finite interval. The authors introduce and characterize quadrature formulas on the unit circle, of Szegö- and interpolatory-type, along with certain error bounds requiring a low computational effort. A method for the computation of integrals on the whole real line with nearby singularities is presented. The procedure consists in the introduction of the possible singularities of the integrand in a weight function defined on \(\mathbb{R}\), considers an associated weight function on the unit circle and then makes use of Szegö- or interpolatory-type quadrature formulas. This procedure is applied in order to provide a computational method for the Fourier transform of a function exhibiting polar singularities near the range of integration. Some error bounds for the estimations are presented and some numerical experiments are carried out.