Error bounds for Gaussian quadrature formulae with Bernstein-Szego weights that are rational modifications of Chebyshev weight functions of the second kind

Abstract

For analytic functions we study the kernel of the remainder terms of Gaussian quadrature rules with respect to Bernstein-Szego weight functions lt graphic xlink:href="drr044eq1" xmlns:xlink="http://www.w3.org/1999/xlink"/> where 0 lt alpha lt beta, beta not equal 2 alpha, vertical bar delta vertical bar lt beta-alpha, and whose denominator is an arbitrary polynomial of exact degree 2 that remains positive on [-1, 1]. The subcase alpha=1, beta=2/(1+gamma), -1 lt gamma lt 0 and delta=0 has been considered recently by Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057). The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds for the corresponding Gaussian quadratures. The approach we use in this paper, which is different from the one adopted in Spalevic, M. M. & Pranic, M. S. ((2010) Error bounds of certain Gaussian quadrature formulae. J. Comput. Appl. Math., 234, 1049-1057), ensures that the actual conditions for determining the locations on the elliptic contours where the modulus of the kernel attains its maximum value are approximated very precisely.