On the 2-Condition Number of Infinite Hankel Matrices of Finite Rank
F. S. V. Bazan and Ph. L. Toint
Report 98/10
Let H be an infinite Hankel matrix whose (i,j)-entry is h_{i+j-2}, where
{h_k}_{k=0}^{\infty} denotes a complex-valued sampled signal: h_k =
\sum_{l=1}^{n} r_l\,z_l^{k}$, $\;|z_l|< 1. Then H has rank n and can be
factorized as H = WRW^T, where W is an infinite Vandermonde matrix in
\C^{\infty\times n} with z_{i}^{j-1} as its (i,j)-entry and R a diagonal
matrix containing the weights r_l. We derive upper bounds for the 2-condition
number of H as functions of n, r_l and z_l, which show that this condition
number is small whenever the z's are close to the unit circle but not
extremely close to each other. An application in problems related to
exponential modeling and system identification is presented and discussed.