A CHARACTERIZATION OF POSITIVE QUADRATURE-FORMULAS

A positive quadrature formula with n nodes which is exact for polynomials of degree 2n - r - 1 , 0 less-than-or-equal-to r less-than-or-equal-to 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.