Optimal Dynamical Frames

TL;DR

This paper characterizes operators generating Parseval dynamical frames in infinite-dimensional Hilbert spaces, introduces the frame index concept, and explores conditions for frames generated by operator iterations.

Contribution

It provides necessary and sufficient conditions for operators to generate Parseval dynamical frames and introduces the concept of frame index with explicit formulas.

Findings

Characterization of operators generating Parseval dynamical frames

Introduction of the frame index for operators

Equivalence of frame indices for T and T* when both admit frames

Abstract

Motivated by the dynamical sampling problem, we study frames in an infinite dimensional Hilbert space generated by the iterates of a bounded operator T, also known as dynamical frames. We first characterize the operators that generate Parseval dynamical frames by showing that the previously known sufficient conditions for their existence are also necessary. We then introduce the frame index of T, the minimal number of vectors required to generate a frame by iterations, and derive an explicit formula for it in the Parseval case together with a general condition for the non-Parseval setting. Finally, we prove that if both $T$ and $T^*$ admit frames of iterations, then their frame indices coincide through an explicit construction.