Dynamical Sampling: A Survey

TL;DR

This survey reviews the theoretical foundations, recent advances, and future directions in dynamical sampling, a mathematical framework for recovering information from space-time samples of evolving signals.

Contribution

It provides a comprehensive overview of the mathematical theory, summarizes recent results, and discusses open problems and future research directions in dynamical sampling.

Findings

Summarized key theoretical results in dynamical sampling.

Identified open problems and conjectures for future research.

Connected dynamical sampling to various mathematical disciplines.

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.