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.