Quadratures and orthogonality associated with the Cayley transform

Abstract

In this paper, we are dealing with the approximate calculation of weighted integrals over the whole real line. The method is based in passing to the unit circle by means of the so-called “Cayley transform”, $z=\frac{i-x}{i+x}$ and then making use of a Szegő or interpolatory-type quadrature formula on the unit circle, in order to obtain a Gauss-like quadrature rule on the real line. Some properties concerning orthogonality, maximal domains of validity of the quadratures and connections with certain orthogonal rational functions are presented. Finally, some numerical experiments are also carried out.