A characterization of positive quadrature formulae

A positive quadrature formula with n nodes which is exact for polynomials of degree In — r — 1, 0 < r < « , 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.