A characterization of positive quadrature formulae

Abstract

A positive quadrature formula with n nodes which is exact for polynomials of degree 2 n − r − 1 , 0 ≤ r ≤ n 2n - r - 1,0 \leq r \leq n , is based on the zeros of certain quasi-orthogonal polynomials of degree n. We show that the quasi-orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries. As a consequence, for a fixed n, every positive quadrature formula is a Gaussian quadrature formula for some nonnegative measure.