Error bounds for Gauss-type quadratures with Bernstein–Szegő weights

Abstract

The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein−Szegő weights,

$${\int\limits_{-1}^{1}}f(t)w(t)\, dt=G_{n}[f]+R_{n}(f),\quad G_{n}[f]=\sum\limits_{\nu=1}^{n}\lambda_{\nu} f(\tau_{\nu}) \quad(n\in\textbf{N}),$$

where f is an analytic function inside an elliptical contour $\mathcal{E}_{\rho}$ with foci at $\mp 1$ and sum of semi-axes $\rho > 1$, and w is a nonnegative and integrable weight function of Bernstein−Szegő type. The derivation of effective bounds on $|R_{n}(f)|$ is possible if good estimates of $\max_{z\in\mathcal{E}_{\rho}}|K_{n}(z)|$ are available, especially if one knows the location of the extremal point $\eta\in\mathcal{E}_{\rho}$ at which $|K_{n}|$ attains its maximum. In such a case, instead of looking for upper bounds on $\max_{z\in\mathcal{E}_{\rho}}|K_{n}(z)|$, one can simply try to calculate $|K_{n}(\eta,w)|$. In the case under consideration, i.e. when

$$w(t)= \frac{(1-t^{2})^{-1/2}}{\beta(\beta-2\alpha)\,t^{2} +2\delta(\beta-\alpha)\,t+\alpha^{2}+\delta^{2}},\quad t\in(-1,1),$$

for some $\alpha,\beta,\delta$, which satisfy $0<\alpha<\beta,\ \beta\ne 2\alpha,\vert\delta\vert<\beta-\alpha$, the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on $|R_{n}(f)|$. The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands.