Szegö polynomials: Quadrature rules on the unit circle and on \([-1,1]\)
Instead of the classical Joukwoski transformation mapping the unit circle onto the interval \([-1,1]\), in this paper, the authors make use of the transformation introduced by \textit{P. Delsarte} and \textit{Y. V. Genin} [IEEE Trans. Acoust. Speech Signal Process. 34, 470--478 (1986)], \[ x= x(z)= (z^{1/2}+ z^{-1/2})/2 \] in order to establish some connections between the families of orthogonal polynomials on \([-1,1]\) with respect to two given positive measures \(dp^{(1)}(x)\) and \(dp^{(2)}(x)\) such that \(dp^{(1)}(x)= (1-x^2)dp^{(2)}(x)\) and the Szegő polynomials on the unit circle with respect to the Borel measure \(dm(z)=- dp^{(1)}(x(z))\). Connection is made through the so-called para-orthogonal polynomials. Relations between certain interpolatory quadrature rules with positive weights or coefficients to estimate integrals both on \([-1,1]\) and the unit circle are also investigated.
scientific article; zbMATH DE number 2060093