Gauss–Kronrod quadrature formulae for weight functions of Bernstein–Szegö type

Abstract

We study the Kronrod extensions of Gaussian quadrature rules whose weight functions on [−1, 1] consist of any one of the four Chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on [−1, 1]. We show that in almost all cases these extended “Gauss–Kronrod” quadrature rules have all the desirable properties: Kronrod nodes interlacing with Gauss nodes, all nodes contained in [−1, 1], and all weights positive and representable by semiexplicit formulas. Exceptions to these properties occur only for small values of n (the number of Gauss nodes), namely n ⩽ 3, and are carefully identified. The precise degree of exactness of each of these Gauss–Kronrod formulae is determined and shown to grow like 4n, rather than 3n, as is normally the case. Our findings are the result of a detailed analysis of the underlying orthogonal polynomials and “Stieltjes polynomials”. The paper concludes with a study of the limit case of a linear divisor polynomial in the weight function.