Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kadec-type theorems for sampled group orbits

Ilya Krishtal, Brendan Miller

TL;DR

The paper generalizes Kadec's $1/4$ perturbation theorem to frames and Riesz bases formed by sampling an orbit $\\{\\mathcal{T}(\\gamma_n)x\\}$ under an isometric representation with Beurling spectrum $\\Lambda(\\mathcal{H},\\mathcal{T})\\subseteq[-\\gamma,\\gamma]$. It develops the $\\mathcal{F}L^1_{loc}$ operator functional calculus for generators of Banach $L^1(\\mathbb{R})$-modules and uses a Paley–Wiener–type perturbation lemma together with a Kadec-style decomposition to derive explicit perturbation bounds on frame constants under $\\delta=\\sup_n|\\tilde{\\gamma}_n-\\gamma_n|$; in particular, if $\\delta$ is below the threshold, the perturbed system preserves frame (and Riesz-basis) properties with computable new bounds. The results extend to atomic decompositions in Banach spaces via Christensen– Heil, providing analogous bounds and basis preservation under perturbation. The work broadens Kadec-type perturbation theory to dynamical sampling contexts and suggests future extensions to non-isometric representations and related theorems (Katsnelson, Avdonin).

Abstract

We extend the classical Kadec 1/4 theorem for systems of exponential functions on an interval to frames and atomic decompositions formed by sampling an orbit of a vector under an isometric group representation.

Kadec-type theorems for sampled group orbits

TL;DR

The paper generalizes Kadec's perturbation theorem to frames and Riesz bases formed by sampling an orbit under an isometric representation with Beurling spectrum . It develops the operator functional calculus for generators of Banach -modules and uses a Paley–Wiener–type perturbation lemma together with a Kadec-style decomposition to derive explicit perturbation bounds on frame constants under ; in particular, if is below the threshold, the perturbed system preserves frame (and Riesz-basis) properties with computable new bounds. The results extend to atomic decompositions in Banach spaces via Christensen– Heil, providing analogous bounds and basis preservation under perturbation. The work broadens Kadec-type perturbation theory to dynamical sampling contexts and suggests future extensions to non-isometric representations and related theorems (Katsnelson, Avdonin).

Abstract

We extend the classical Kadec 1/4 theorem for systems of exponential functions on an interval to frames and atomic decompositions formed by sampling an orbit of a vector under an isometric group representation.
Paper Structure (5 sections, 7 theorems, 30 equations)

This paper contains 5 sections, 7 theorems, 30 equations.

Table of Contents

  1. Introduction and main results
  2. $\mathcal{F}{L^1_{loc}}$ functional calculus
  3. Proofs of the main results.
  4. Extension to atomic decompositions
  5. Concluding remarks

Key Result

Theorem 1.1

Let $\mathcal{T}:{\mathbb R}\to B({\mathcal{H}})$ be an isometric representation such that $\Lambda({\mathcal{H}},\mathcal{T}) \subseteq [-\gamma,\gamma]$ for some $\gamma > 0$. Assume that a vector $x\in{\mathcal{H}}$ and a set $\Gamma = \{\gamma_n: n\in{\mathbb Z}\} \subset{\mathbb R}$ are such th Then the system of vectors $\{\mathcal{T}(\widetilde{\gamma}_n)x\}$ also forms a frame for ${\mathc

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['Bineq']}
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.1: C95
  • ...and 4 more