# The Generalized Operator Based Prony Method

## TL;DR

This paper extends the generalized Prony method for sparse signal reconstruction by introducing operator transformations and new sampling functionals to simplify computations and enhance numerical stability.

## Contribution

It proposes two extensions: transforming the operator with an analytic function and selecting new sampling functionals to reduce computational complexity.

## Key findings

- Transforming operators simplifies power evaluations.
- New sampling functionals reduce the number of required operator powers.
- Extensions improve numerical stability and efficiency.

## Abstract

The generalized Prony method introduced by Peter & Plonka (2013) is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator $A$. However, this procedure requires the evaluation of higher powers of the linear operator $A$ that are often expensive to provide. In this paper we propose two important extensions of the generalized Prony method that simplify the acquisition of the needed samples essentially and at the same time can improve the numerical stability of the method. The first extension regards the change of operators from $A$ to $\varphi(A)$, where $\varphi$ is an analytic function, while $A$ and $\varphi(A)$ possess the same set of eigenfunctions. The goal is now to choose $\varphi$ such that the powers of $\varphi(A)$ are much simpler to evaluate than the powers of $A$. The second extension concerns the choice of the sampling functionals. We show, how new sets of different sampling functionals $F_{k}$ can be applied with the goal to reduce the needed number of powers of the operator $A$ (resp. $\varphi(A)$) in the sampling scheme and to simplify the acquisition process for the recovery method.