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In this paper, we consider the Gauss-Kronrod quadrature formulas for a modified Chebyshev weight. Efficient estimates of the error of these Gauss–Kronrod formulae for analytic functions are obtained, using techniques of contour integration that were introduced by Gautschi and Varga (cf. Gautschi and Varga SIAM J. Numer. Anal. 20, 1170–1186 1983). Some illustrative numerical examples which show both the accuracy of the Gauss–Kronrod formulas and the sharpness of our estimations are displayed. Though for the sake of brevity we restrict ourselves to the first kind Chebyshev weight, a similar analysis may be carried out for the other three Chebyshev type weights; part of the corresponding computations are included in a final appendix. [ABSTRACT FROM AUTHOR]

Abstract: Micchelli and Sharma constructed in their paper [On a problem of Turán: multiple node Gaussian quadrature, Rend. Mat. 3 (1983) 529–552] a quadrature formula for the Fourier–Chebyshev coefficients, which has the highest possible precision. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points and a sum of semi-axes , for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the location on the ellipses where the maximum modulus of the kernel is attained, and derive effective error bounds for this quadrature formula. Numerical examples are included. [Copyright &y& Elsevier]

Abstract: For analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points and a sum of semi-axes , for Gauss–Radau quadrature formula with Chebyshev weight function of the third kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi in his paper [W. Gautschi, On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures, Rocky Mounatin J. Math. 21 (1991) 209-206]. In this way the last unproved conjecture from the mentioned paper is now verified. [Copyright &y& Elsevier]

In this paper, we consider the Gauss quadrature formulae corresponding to some modifications of each of the four Chebyshev weights, considered by Gautschi and Li in [4]. As it is well known, in the case of analytic integrands the error of these quadrature formulas can be represented as a contour integral with a complex kernel. We study the kernel of the mentioned quadrature formulas on suitable elliptic contours, in such a way that the behavior of its modulus is analyzed in a rather simple manner, allowing us to derive some effective error bounds. In addition, some numerical examples checking the accuracy of such error bounds are included. [ABSTRACT FROM AUTHOR]

Abstract: We continue with the study of the kernels K n (z) in the remainder terms R n (f) of the Gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at ∓1 and a sum of semi-axes ρ >1. The weight function w of Bernstein–Szegő type here isSufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis. This leads to effective error bounds of the corresponding Gauss quadratures. The quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands. [Copyright &y& Elsevier]