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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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##### References:
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