Linear Combinations of Orthogonal Polynomials Generating Positive Quadrature Formulas

Abstract

Let ${p_k}(x) = {x^k} + \cdots$, $k \in {{\mathbf {N}}_0}$, be the polynomials orthogonal on $[ - 1, + 1]$ with respect to the positive measure $d\sigma$. We give sufficient conditions on the real numbers ${\mu _j}$, $j = 0, \ldots ,m$, such that the linear combination of orthogonal polynomials $\sum _{j = 0}^m{\mu _j}{p_{n - j}}$ has n simple zeros in $( - 1, + 1)$ and that the interpolatory quadrature formula whose nodes are the zeros of $\sum _{j = 0}^m{\mu _j}{p_{n - j}}$ has positive weights.