Stieltjes polynomials and related quadrature formulae for a class of weight functions

Abstract:

Consider a (nonnegative) measure $d \sigma$ with support in the interval $[a,b]$ such that the respective orthogonal polynomials, above a specific index $\ell$, satisfy a three-term recurrence relation with constant coefficients. We show that the corresponding Stieltjes polynomials, above the index $2\ell -1$, have a very simple and useful representation in terms of the orthogonal polynomials. As a result of this, the Gauss-Kronrod quadrature formulae for $d \sigma$ have all the desirable properties, namely, the interlacing of nodes, their inclusion in the closed interval $[a,b]$ (under an additional assumption on $d \sigma$), and the positivity of all weights. Furthermore, the interpolatory quadrature formulae based on the zeros of the Stieltjes polynomials have positive weights, and both of these quadrature formulae have elevated degrees of exactness.