The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegö type
We consider the Gaussian quadrature formulae \(\int^{1}_{- 1}f(t)w(t)dt=\sum^{n}_{\nu =1}\lambda_{\nu}f(\tau_{\nu})+R_ n(f)\) relative to the weight functions \(w(t)=(1- t)^{\alpha}(1+t)^{\beta}/\rho (t)\), \(-1<t<1\), \(\alpha =\pm\), \(\beta =\pm\), where \(\rho\) (t) is an arbitrary quadratic polynomial which remains positive on [-1,1]. If \(f(z)=\sum^{\infty}_{k=0}a_ kz^ k\) is a holomorphic function in \(C_ r=\{z\in {\mathbb{C}}:| z| <r\}\), \(r>1\), we define \(| f|_ r=\sup \{| a_ k| r^ k:k\in {\mathbb{N}}_ 0\) and \(R_ n(t^ k)\neq 0\}\), and we set \(X_ r=\{f:f\) holomorphic in \(C_ r\) and \(| f|_ r<\infty \}\). Then in \((X_ r,| \cdot |_ r)\), the error term \(R_ n\) is a continuous linear functional. In the present paper, using the method of \textit{G. Akrivis} [Math. Comput. 45, 513-519 (1985; Zbl 0602.41030)], we explicitly compute \(\| R_ n\|\) for the quadrature formulae in consideration. This subsequently leads to bounds for \(R_ n\), of the form \(| R_ n(f)| \leq \| R_ n\| | f|_ r\). The quality of these bounds is demonstrated by two numerical examples.
scientific article; zbMATH DE number 4107376