In this paper we present a procedure for the computation of integrals on the whole real line with nearby singularities. The method is based in collecting the possible singularities of the integrand within a weight function on the real line, passing to the unit circle by considering an associated weight function there and then making use of Szego or interpolatory-type quadrature formulas. This algorithm 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 carried out.

This work is partially supported by Dirección General de Programas y Transferencia de Conocimiento, Ministerio de Ciencia e Innovación of Spain under grant MTM 2008-06671.
The work of the third author has been partially supported also by a Grant of Agencia Canaria de Investigación, Innovación y Sociedad de la Información del Gobierno de Canarias.