Application of the AAK theory for sparse approximation of exponential sums

TL;DR

This paper introduces a novel method combining Prony's method and AAK-theory to efficiently approximate discrete signals as short exponential sums, improving sparse representation accuracy.

Contribution

It adapts AAK-theory for sparse exponential sum approximation of discrete signals, providing a new algorithm with theoretical guarantees and practical effectiveness.

Findings

The algorithm accurately recovers exponential sums from signals.

Numerical tests demonstrate improved approximation quality.

The method is grounded in linear algebra and Fourier analysis.

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.