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Optimal Dynamical Frames

A. Aguilera, C. Cabrelli, F. Negreira, V. Paternostro

TL;DR

The paper addresses when frames generated by operator iteratees, or dynamical frames, exist in infinite-dimensional Hilbert spaces and how to minimize generators. It establishes a complete Parseval characterization: a Parseval dynamical frame exists iff the operator is a contraction with a strongly stable adjoint, and it provides an explicit Parseval frame index $\gamma_p(T)=\dim \overline{(I- TT^*)(\mathcal H)}$; for general frames, $\gamma(T)$ equals the minimum Parseval index over contractions similar to $T$. The authors develop a model-space framework linking frames of iterations to shift compressions, prove joint strong stability equivalences with full-range subspaces, and construct optimal Parseval frames via these models, showing $\gamma_p(T)=\gamma_p(T^*)$ under joint stability and providing concrete procedures for generating frames. The results yield constructive methods for dynamical frames with minimal, independent generators and illuminate when frames for $T$ and $T^*$ are similar, leveraging Beurling-Lax-Halmos theory and inner-function factorization. These insights advance dynamical sampling theory and operator-frame theory by giving precise, transferable criteria and explicit constructions in a robust functional-model setting.

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.

Optimal Dynamical Frames

TL;DR

The paper addresses when frames generated by operator iteratees, or dynamical frames, exist in infinite-dimensional Hilbert spaces and how to minimize generators. It establishes a complete Parseval characterization: a Parseval dynamical frame exists iff the operator is a contraction with a strongly stable adjoint, and it provides an explicit Parseval frame index ; for general frames, equals the minimum Parseval index over contractions similar to . The authors develop a model-space framework linking frames of iterations to shift compressions, prove joint strong stability equivalences with full-range subspaces, and construct optimal Parseval frames via these models, showing under joint stability and providing concrete procedures for generating frames. The results yield constructive methods for dynamical frames with minimal, independent generators and illuminate when frames for and are similar, leveraging Beurling-Lax-Halmos theory and inner-function factorization. These insights advance dynamical sampling theory and operator-frame theory by giving precise, transferable criteria and explicit constructions in a robust functional-model setting.

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 and admit frames of iterations, then their frame indices coincide through an explicit construction.

Paper Structure

This paper contains 19 sections, 45 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper and main results
  3. Preliminares
  4. Model spaces
  5. Basic frames
  6. Existence of frames of iterations
  7. Examples of frames of iterations
  8. Frame Index of a bounded operator
  9. Frame indexes
  10. Linear independence of generators
  11. Strongly Stable Contractions with Strongly Stable Adjoints
  12. Full range subspaces and inner functions
  13. Strong stability of the shift compression and full range subspaces
  14. Optimal Parseval frames of iterations
  15. Optimal Parseval frames in model spaces
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $T\in{\mathcal{B}}({\mathcal{H}})$. There exists a set of vectors $\{v_i\}_{i\in I}\subseteq{\mathcal{H}}$ such that $\{T^nv_i\}_{i\in I,n\geqslant0}$ is a Parseval frame in ${\mathcal{H}}$ if and only if ${\left\|{T}\right\|}\leqslant1$ and $(T^*)^nv\to0$ as $n\to+\infty$ for all $v\in{\mathcal

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2: CMPP20
  • Theorem 1.3
  • Proposition 2.1: ACCP
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • ...and 75 more