Gaussian quadrature rules with multiple nodes with respect to orthogonality on the semicircle

Abstract

Weighted orthogonality on the semicircle was introduced in 1987 in \cite{OP2}. In that paper, orthogonality is considered with respect to a non-Hermitian inner product defined by \[ (f,g)=\int\limits_{\Gamma}f(z)g(z)(\mathrm{i}z)^{-1}w(z)\mathrm{d}z=\int\limits_0^{\pi} f(\mathrm{e}^{\mathrm{i}\theta})g(\mathrm{e}^{\mathrm{i}\theta})w(\mathrm{e}^{\mathrm{i}\theta})\mathrm{d}\theta, \] where $\Gamma=\{z=\mathrm{e}^{\mathrm{i}\theta}\, :\, 0\leq \theta \leq \pi\}$. Gaussian quadrature rules associated with orthogonality on the semicircle were introduced in [2]. On the other hand, on the real line, Gaussian quadrature rules with multiple nodes have been studied in [3] and [4]. These are the so-called Gauss-Tur\'{a}n quadrature rules, defined with respect to $s$-orthogonality, and the Chakalov-Popoviciu rules, defined with respect to $\sigma$-orthogonality. We introduce Gaussian quadrature rules with respect to $s$-orthogonality and $\sigma$-orthogonality on the semicircle.