  Matrices with displacement structure such as Pick, Vandermonde, and Hankel
matrices appear in a diverse range of applications. In this paper, we use an
extremal problem involving rational functions to derive explicit bounds on the
singular values of such matrices. For example, we show that the $k$th singular
value of a real $n\times n$ positive definite Hankel matrix, $H_n$, is bounded
by $C\rho^{-k/\log n}\|H\|_2$ with explicitly given constants $C>0$ and
$\rho>1$, where $\|H_n\|_2$ is the spectral norm. This means that a real
$n\times n$ positive definite Hankel matrix can be approximated, up to an
accuracy of $\epsilon\|H_n\|_2$ with $0<\epsilon<1$, by a rank
$\mathcal{O}(\log n\log(1/\epsilon) )$ matrix. Analogous results are obtained
for Pick, Cauchy, real Vandermonde, L\"{o}wner, and certain Krylov matrices.
