Application of the AAK theory for sparse approximation of exponential sums

Abstract

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.

24 pages, 1 figure

Cited by in corpus (1)

• Optimal Rank-1 Hankel Approximation of Matrices: Frobenius Norm, Spectral Norm, and Cadzow's Algorithm