A fast algorithm for generalized Hankel matrices arising in finite-moment problems

Consider an n × n lower triangular matrix L whose (i + 1)st row is defined by the coefficients of the real polynomial pi(x) of degree i such that {pi(x)} is a set of orthogonal polynomials satisfying a standard three-term recurrence relation. If H is an n × n real Hankel matrix with nonsingular leading principal submatrices, then will be referred to as a strongly nonsingular Hankel matrix with respect to the orthogonal polynomial basis {pi(x)}. In this paper we develop an efficient O(n2) algorithm for the solution of a system of linear equations with a real symmetric coefficient matrix which is a Hankel matrix with respect to a suitable orthogonal polynomial basis. This leads to an efficient method for computing polynomial approximations of an unknown function given its modified moments.