Eigenvalue estimates for Hankel matrices

Positive-definite Hankel matrices have an important property: the ratio of the largest and the smallest eigenvalues (the spectral condition number) has as a lower bound an increasing exponential of the order of the matrix that is independent of the particular matrix entries. The proof of this fact is related to the so-called Vandermonde factorizations of positive-definite Hankel matrices. In this paper the structure of these factorizations is studied for real sign-indefinite strongly regular Hankel matrices. Some generalizations of the estimates of the spectral condition number are suggested.