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Prony's Method in Banach Modules

Ilya A. Krishtal, Götz E. Pfander

Abstract

We show that the classical Prony's method for recovery of a sparse signal from its consecutive Fourier coefficients can be viewed as a spectral identification problem for an unknown restriction of a known linear operator. This presents a unified point of view on various existing and novel generalizations and applications of the method, some of which are discussed in this paper.

Prony's Method in Banach Modules

Abstract

We show that the classical Prony's method for recovery of a sparse signal from its consecutive Fourier coefficients can be viewed as a spectral identification problem for an unknown restriction of a known linear operator. This presents a unified point of view on various existing and novel generalizations and applications of the method, some of which are discussed in this paper.
Paper Structure (8 sections, 8 theorems, 52 equations, 1 algorithm)

This paper contains 8 sections, 8 theorems, 52 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Banach modules and spectral identification
  3. Reduction to Prony's method.
  4. Standard Prony's method.
  5. Exponential sums with polynomial coefficients.
  6. Sparse Dynamical Sampling
  7. Identification of time-varying channels
  8. Concluding remarks

Key Result

Theorem 2.1

The spectrum $\sigma(B_F)$ of the operator $B_F$ satisfies

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Remark 2.1
  • Theorem 2.1: B04
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.3
  • Definition 2.4
  • ...and 15 more