Convergent interpolatory quadrature rules and orthogonal polynomials of varying measures

Abstract

Let (P _n ) be a sequence of polynomials such that P _n (x) > 0 for x ∈ [− 1, 1] and $\lim \limits _{n\to \infty }\text {deg}(P_{n})/n = 1$. Let q _n be the nth monic orthogonal polynomial with respect to ${P}_{n}^{-1}$ d μ, where μ is a measure on [− 1, 1] that is regular in the sense of Stahl and Totik. We prove that the interpolatory quadrature rule with nodes at the zeros of q _n is convergent with respect to μ provided that the zeros of P _n lie outside a certain curve surrounding [− 1, 1].