The remainder term for analytic functions of symmetric Gaussian quadratures

Abstract

Abstract. For analytic functions the remainder term of Gaussian quadrature rules can be expressed as a contour integral with kernel Kn. In this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially Gegenbauer weight functions. First a new series representation of the kernel is developed and analyzed. Then the location of the maximum modulus of the kernel on suitable ellipses is determined. Depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis. Finally, a detailed discussion for some special weight functions is given.

Error bounds for Gauss-Turán quadrature formulae of analytic functions

“…In particular, they investigated some cases with special Jacobi weights with parameters ±1/2 (Chebyshev weights). The cases of Gaussian rules with Bernstein-Szegő weight functions and with some symmetric weights including especially the Gegenbauer weight were studied by Peherstorfer [35] and Schira [38], respectively. Some of the results have been extended to Gauss-Radau and Gauss-Lobatto formulas (cf.…”

On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions

“…We note that the special cases σ = 0 and σ = 1 of Theorem 1.1 were already proved by Schira (see [10,Theorem 3.1] and [8, (4.13…”

“…The kernel K G n (·; w) of the Gauss quadrature formula (1.10) has been investigated on elliptical contours in a recent paper of T. Schira [10].…”

Maximum of the modulus of kernels in Gauss-Turán quadratures

“…They applied it to several kinds of interpolatory and non-interpolatory quadrature rules. Error bounds for Gaussian quadratures of analytic functions were studied by Gautschi and Varga [5] (see also [6]), and later by Schira [15,16], Hunter and Nikolov [8].…”