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The error norm of Gaussian quadrature formulae for weight functions of Bernstein-Szegö type. (English) Zbl 0676.41034
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 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.
Reviewer: S.E.Notaris