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Dynamical Sampling: A Survey

Akram Aldroubi, Carlos Cabrelli, Ilya Krishtal, Ursula Molter

TL;DR

Dynamical sampling addresses how to recover quantities of interest for time-evolving signals from space–time measurements by studying frames generated by operator iterates. The survey draws deep connections between frames of iterations, operator theory, and Hardy-space models to cover discrete and continuous-time settings, finite and infinite dimensions, and homogeneous and inhomogeneous dynamics. It catalogs complete frame-characterization results in finite dimensions, analyzes the existence and structure of frames in infinite dimensions (including contractions and Carleson-type phenomena), and presents functional-model approaches that transfer operator questions to analytic spaces. The work also surveys source recovery and system identification within this framework, and outlines open problems and future directions across discretization, design of sensor sets, and extensions to graphs and Banach spaces.

Abstract

Dynamical sampling refers to a class of problems in which space-time samples are taken from a signal evolving under an underlying dynamical system. The goal is to use these samples to recover relevant information about the system, such as the initial state, the evolution operator, or the sources and sinks driving the dynamics. These problems are tightly connected to frame theory, operator theory, functional analysis, and other foundational areas of mathematics; they also give rise to new theoretical questions and have applications across engineering and the sciences. This survey provides an overview of the theoretical underpinnings of dynamical sampling, summarizes recent results, and outlines directions for future work, including open problems and conjectures.

Dynamical Sampling: A Survey

TL;DR

Dynamical sampling addresses how to recover quantities of interest for time-evolving signals from space–time measurements by studying frames generated by operator iterates. The survey draws deep connections between frames of iterations, operator theory, and Hardy-space models to cover discrete and continuous-time settings, finite and infinite dimensions, and homogeneous and inhomogeneous dynamics. It catalogs complete frame-characterization results in finite dimensions, analyzes the existence and structure of frames in infinite dimensions (including contractions and Carleson-type phenomena), and presents functional-model approaches that transfer operator questions to analytic spaces. The work also surveys source recovery and system identification within this framework, and outlines open problems and future directions across discretization, design of sensor sets, and extensions to graphs and Banach spaces.

Abstract

Dynamical sampling refers to a class of problems in which space-time samples are taken from a signal evolving under an underlying dynamical system. The goal is to use these samples to recover relevant information about the system, such as the initial state, the evolution operator, or the sources and sinks driving the dynamics. These problems are tightly connected to frame theory, operator theory, functional analysis, and other foundational areas of mathematics; they also give rise to new theoretical questions and have applications across engineering and the sciences. This survey provides an overview of the theoretical underpinnings of dynamical sampling, summarizes recent results, and outlines directions for future work, including open problems and conjectures.

Paper Structure

This paper contains 18 sections, 16 theorems, 43 equations.

Table of Contents

  1. Introduction and Motivation
  2. Space-time trade-off and frames of iterations
  3. Discrete case
  4. Discrete dynamical frames. Finite dimensional case
  5. Diagonalizable Transformations
  6. General linear transformations
  7. Discrete dynamical frames. Infinite dimensional case
  8. Normal operators
  9. General operators
  10. Frame-index of an operator
  11. Set of generators
  12. Continuous-time case: formulation and semi-continuous frames
  13. Convolution-kernel (semigroup) case
  14. Source recovery and frames
  15. Discrete-time model for source recovery
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\{b_i: i \in J\}$ be vectors in $\mathbb{C}^d$. Let $D$ be a diagonal matrix and $r_i$ the degree of the $D$-annihilator of $b_i$. Set $\ell_i=r_i-1$. Then $\{D^{j}b_i: \; i\in J, \, j=0, \dots, \ell_i\}$ is a frame of $\mathbb{C}^d$ if and only if $\{P_j(b_i):i \in \Omega\}$ form a frame of $

Theorems & Definitions (21)

  • Theorem 2.1: ACMT17
  • Theorem 2.2: ACMT17, Theorem 2.6
  • Theorem 2.3: AP17
  • Definition 2.4: Contraction and strong stability
  • Definition 2.5: Similarity
  • Theorem 2.6: CMPP20ACNP25
  • Theorem 2.7: ACNP25
  • Theorem 2.8: ACMT17ACCMP17AP17
  • Theorem 2.9
  • Conjecture 1
  • ...and 11 more