Prony methods for recovery of structured functions

Abstract

In this survey, we describe the classical Prony method and whose relatives. We sketch a frequently used Prony–like method for equispaced sampled data, namely the ESPRIT method. The case of nonequispaced sampled data is discussed too. For the reconstruction of a sparse eigenfunction expansion, a generalized Prony method is presented. The Prony methods are applied to the recovery of structured functions (such as exponential sums and extended exponential sums) and of sparse vectors. The recovery of spline functions with arbitrary knots from Fourier data is also based on Prony methods. Finally, some numerical examples are given. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interpolation by generalized exponential sums with equal weights

“…Furthermore, in spite of the huge bibliography related to the Prony method and general amplitude and frequency sums (see [2,3,5,12,20,23,24] and references therein), we could not find any more or less general estimates for amplitudes and frequencies similar to those in Theorem 5. Probably, they just do not exist because of the above-mentioned divergence examples from [12, Section 7] and the results from [2].…”

The Generalized Operator Based Prony Method

“…, 2M − 1, exact recovery is possible if T j ∈ R + i[−π, π), see e.g. [18]. Usually, we assume that there is an a priori known bound C such that Im T j ∈ [−Cπ, Cπ), and the parameters T j can still be recovered using a rescaling argument and taking sampling values f (kh) with h ≤ 1/C instead of for the vector p = (p 0 , .…”

Sparse Phase Retrieval of One-Dimensional Signals by Prony's Method

“…However, at this stage, the bijective mapping between ℓ > 0 and (j, k) with j > k such that τ ℓ = T j − T k will be still unknown and needs to be found in a second reconstruction step. In order to recover the frequency differences τ ℓ and the unknown coefficients γ ℓ from the exponential sum (3.2) we employ Prony's method [24,25].…”