# A Fast Algorithm for Hankel Matrices Represented in Orthogonal Polynomial Bases

Consider a $n \times n$ lower triangular matrix $L$ whose
$(i+1)$-st row is defined by the coefficients of the real
polynomial $p_i(x)$ of degree $i$ such that
$\{ p_i(x)\}$ is a set of orthogonal polynomials
satisfying a standard three-term
recurrence relation.
If $H$ is a $n\times n$ real Hankel matrix with nonsingular
leading principal submatrices, then $\widehat{H}=L HL^T$ will be referred
as a strongly nonsingular Hankel matrix with respect to the
orthogonal polynomial basis $\{p_i(x)\}$.
In this paper we develop an efficient
$O(n^2)$ algorithm for the solution of a system of linear equations with a
real symmetric coefficient matrix $\widehat{H}$ which is a Hankel matrix
with respect to a suitable orthogonal polynomial basis. We then apply our
method to the solution of a weighted
finite moment problem.