  In this paper, we derive a new method for optimal $\ell^{1}$- and
$\ell^2$-approximation of discrete signals on ${\mathbb N}_{0}$ whose entries
can be represented as an exponential sum of finite length. Our approach employs
Prony's method in a first step to recover the exponential sum that is
determined by the signal. In the second step we use the AAK-theory to derive an
algorithm for computing a shorter exponential sum that approximates the
original signal in the $\ell^{p}$-norm well. AAK-theory originally determines
best approximations of bounded periodic functions in Hardy-subspaces. We
rewrite these ideas for our purposes and give a proof of the used AAK theorem
based only on basic tools from linear algebra and Fourier analysis. The new
algorithm is tested numerically in different examples.
