TITLE: Eigenvalues of partial Hankel matrices
QUESTION [6 upvotes]: I was wondering if there are closed formulas for the singularvalues of a partial Hankel matrix (by partial I mean $\ell<n$)
\begin{align*}
H=
\begin{bmatrix}
c_1 & c_2 & \ldots & c_\ell \\ c_2 & c_3 & \ldots & c_{\ell+1} \\ 
c_3 & c_4 & \ldots & c_{\ell+2}\\
\vdots & \vdots & \ddots & \vdots\\
c_n & c_{n+1} &\ldots & c_{n+\ell-1}
\end{bmatrix}
\end{align*}
What does it mean for this matrix to be low-rank?

REPLY [2 votes]: Typically, one does "Prony method": considers an infinite (or just long enough) sequence $c=(c_1,c_2,\dots)$ and a system of equations of the form $V(x)Z=c$, with $Z$ a vector of non-0 unknowns, and $V(x)$ the Vandermonde matrix
$$
V(x)=\begin{pmatrix} 
1&1&\dots&1\\
x_1& x_2&\dots &x_k\\
x_1^2& x_2^2&\dots &x_k^2\\
&\dots&\dots&\dots\\
\end{pmatrix} 
$$
By multiplying both sides of $V(x)Z=c$ on the left by the row vectors $a_\ell=(\underbrace{0,\dots,0}_\ell,a_0,a_1,\dots)$, for $\ell\geq 0$, one relates the polynomials $x^\ell a(x)=x^\ell\sum_{i\geq 0} a_i x^i$ with zeros $\{x_1,\dots,x_k\}$ and the kernel of a submatrix of the the infinite Hankel matrix corresponding to the sequence $c$. 
By controlling the degree of $a(x)=k$, you should be able to get something interesting...