Solvability of the Hankel Determinant Problem for Real Sequences

Abstract

To each nonzero sequence $s:= \{s_{n}\}_{n \geq 0}$ of real numbers we associate the Hankel determinants $D_{n} = \det \mathcal{H}_{n}$ of the Hankel matrices $\mathcal{H}_{n}:= (s_{i + j})_{i, j = 0}^{n}$, $n \geq 0$, and the nonempty set $N_{s}:= \{n \geq 1 \, | \, D_{n-1} \neq 0 \}$. We also define the Hankel determinant polynomials $P_0:=1$, and $P_n$, $n\geq 1$ as the determinant of the Hankel matrix $\mathcal H_n$ modified by replacing the last row by the monomials $1, x, \ldots, x^n$. Clearly $P_n$ is a polynomial of degree at most $n$ and of degree $n$ if and only if $n\in N_s $. Kronecker established in 1881 that if $N_s $ is finite then rank $\mathcal{H}_{n} = r$ for each $n \geq r-1$, where $r := \max N_s $. By using an approach suggested by I.S.Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence $\{t_n\}_{n\geq 0}$ to be of the form $t_n=D_n$, $n\geq 0$ for a real sequence $\{s_n\}_{n\geq 0}$. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial $ P_n $ satisfying deg$P_n = n\geq 1$ is preceded by a nonzero polynomial $P_{n-1}$ whose degree can be strictly less than $n-1$ and which has no common zeros with $ P_n $. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that $D_0 > 0, \ldots, D_{r-1} > 0 $ and $D_n=0$ for all $n\geq r$.