We employ the generalized Prony method in [T. Peter and G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators, Inverse Problems 29 (2013) 025001] to derive new reconstruction schemes for a variety of sparse signal models using only a small number of signal measurements. By introducing generalized shift operators, we study the recovery of sparse trigonometric and hyperbolic functions as well as sparse expansions into Gaussians chirps and modulated Gaussian windows. Furthermore, we show how to reconstruct sparse polynomial expansions and sparse non-stationary signals with structured phase functions.

The generalized Prony method introduced in [13] 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 ϕ(A), where ϕ is an analytic function, while A and ϕ(A) possess the same set of eigenfunctions. The goal is now to choose ϕ such that the powers of ϕ(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. ϕ(A)) in the sampling scheme and to simplify the acquisition process for the recovery method.

We derive a method for the reconstruction of non-stationary signals with structured phase functions using only a small number of signal measurements. Our approach employs generalized shift operators as well as the generalized Prony method. Our goal is to reconstruct a variety of sparse signal models using a small number of signal measurements.

In the last three years I had the opportunity to work at the Institute for Mathematical Stochastics at the University of Göttingen, an excellent place for research and learning. I have found colleagues, always ready for discussion about new ideas, which also became friends and learned much about the variety of mathematical research in stochastics as well as in numerics.