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Maximum of the modulus of kernels in Gauss-Turán quadratures. (English) Zbl 1149.41011
Summary: We study the kernels \( K_{n,s}(z)\) in the remainder terms \( R_{n,s}(f)\) of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at \( \pm 1\), when the weight \( \omega\) is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel \( | K_{n,s}(z)|\) attains its maximum on the real axis (positive real semi-axis) for each \( n\geq n_0, n_0=n_0(\rho,s)\). It was stated as a conjecture by the first two authors in [Math. Comput. 72, No. 244, 1855–1872 (2003; Zbl 1030.41018)]. For the generalized Chebyshev weight of the second kind, in the case when the number of the nodes \(n\) in the corresponding Gauss-Turán quadrature formula is even, it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each \( n\geq n_0, n_0=n_0(\rho,s)\). Numerical examples are included.

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