Frames Induced by the Action of Continuous Powers of an Operator

TL;DR

This paper studies systems generated by continuous powers of a normal operator in a Hilbert space, focusing on their frame, completeness, and Bessel properties, with implications for dynamical and mobile sampling in science and engineering.

Contribution

It provides new theoretical insights into the frame and completeness properties of systems formed by continuous operator powers, linking them to sampling problems.

Findings

Analyzes conditions for frame properties of operator power systems

Establishes criteria for completeness and Bessel properties

Connects theoretical results to sampling applications

Abstract

We investigate systems of the form $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$ where $A \in B(\mathcal{H})$ is a normal operator in a separable Hilbert space $\mathcal{H}$, $\mathcal{G}\subset \mathcal{H}$ is a countable set, and $L$ is a positive real number. Although the main goal of this work is to study the frame properties of $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$, as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science.