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Error bounds of certain Gaussian quadrature formulae. (English) Zbl 1205.41034
Let
\[ \int_{-1}^{1} f(t)w(t)\,dt=\sum_{\nu=1}^{n}\lambda_{\nu}f(\tau_{\nu})+R_{n}(f) \tag{1} \]
the Gaussian quadrature formula for analytic functions on elliptical contours with foci \(\mp 1\) and the sum of semi-axes \(\rho>1\),
\[ \varepsilon_{\rho}=\left\{ z\in {\mathbb C}\mid z=\tfrac{1}{2}(\xi+\xi^{-1}),\;0\leq \theta\leq 2\pi\right\}, \quad \xi=\rho e^{i\theta}, \tag{2} \]
where \(w\) is a nonnegative and integrable function on the interval \((-1,1)\), which is exact for all algebraic polynomials of degree at most \(2n-1\). The nodes \(\tau_{\nu}\) in (1) are zeros of the orthogonal polynomials \(\pi_{n}\) with respect to the weight function \(w\).
Here, \(w\) represents the class of symmetric weight functions of Bernstein-Szegő type
\[ w(t)\equiv w_{\gamma}(t)=\frac{\sqrt{1-t^2}}{1-\frac{4\gamma}{(1+\gamma)^2}t^2},\quad t\in(-1,1),\;\gamma\in(-1,0). \]
Let \(\Gamma\) be a simple closed curve in the complex plane surrounding the interval \([-1,1]\) and \({\mathcal D}= \operatorname{int}\Gamma\) its interior. If the integrand \(f\) is analytic in \({\mathcal D}\) and continuous on \(\overline{\mathcal D}\), then the remainder term \( R_{n}(f)\) in (1) admits the contour integral representation
\[ R_{n}(f)=\frac{1}{2\pi i}\oint_{\Gamma} K_{n}(z)f(z)\,dz. \]
The kernel is given by
\[ K_{n}(z)\equiv K_{n}(z,w)=\frac{\rho(z)}{\pi_{n}(z)},\quad z \in[-1,1], \]
where
\[ \rho_{n}(z)\equiv\rho_{n,w}(z)=\int_{-1}^{1}\frac{\pi_{n}(t)}{z-t}w(t)\,dt. \]
The authors study the kernel \(K_{n}(z)\) in the remainder terms \(R_{n}(f)\) of the Gaussian quadrature formula. The location on the elliptic contour, where the modulus of the kernel attains its maximum value, is investigated.

MSC:
41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
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