Application of the AAK theory for sparse approximation of exponential sums

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.