19 research outputs found
Error Estimations of Turan Formulas with Gori-Micchelli and Generalized Chebyshev Weight Functions
S. Li in [Studia Sci. Math. Hungar. 29 (1994) 71-83] proposed a Kronrod type extension to the well-known Turan formula. He showed that such an extension exists for any weight function. For the classical Chebyshev weight function of the first kind, Li found the Kronrod extension of Turan formula that has all its nodes real and belonging to the interval of integration, [-1, 1]. In this paper we show the existence and the uniqueness of the additional two cases - the Kronrod exstensions of corresponding Gauss-Turan quadrature formulas for special case of Gori-Micchelli weight function and for generalized Chebyshev weight function of the second kind, that have all their nodes real and belonging to the integration interval [-1, 1]. Numerical results for the weight coefficients in these cases are presented, while the analytic formulas of the nodes are known
Error bounds of the Micchelli-Sharma quadrature formula for analytic functions
Micchelli and Sharma constructed in their paper [On a problem of Turan: 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 not equal 1 and a sum of semi-axes rho > 1, 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
Error bounds of the Micchelli-Sharma quadrature formula for analytic functions
Micchelli and Sharma constructed in their paper [On a problem of Turan: 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 not equal 1 and a sum of semi-axes rho > 1, 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
Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results
This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses
Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results
This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses
Quadrature Formulas with Multiple Nodes for Fourier Coeffcients
Gaussian quadrature formulas with multiple nodes and their optimal extensions
for computing the Fourier coe cients, in expansions of functions with
respect to a given system of orthogonal polynomials, are considered. A numerically
stable construction of these quadratures is proposed. Error bounds for
these quadrature formulas are derived. We present a survey of recent results on
this topic