An asymptotic equivalence between two frame perturbation theorems

TL;DR

This paper compares two stability theorems for exponential frames, demonstrating their asymptotic equivalence in terms of perturbation bounds as the dimension increases.

Contribution

It proves that two previously known frame perturbation constants are asymptotically equivalent in high dimensions.

Findings

Theorem proven that relates the two stability results.

Constants in the theorems become asymptotically equal as dimension grows.

Provides insight into high-dimensional frame stability behavior.

Abstract

In this paper, two stability results regarding exponential frames are compared. The theorems, (one proven herein, and the other in \cite{SZ}), each give a constant such that if $\sup_{n \in \mathbb{Z^d}}\| \epsilon_n \|_\infty < C$, and $(e^{i \langle \cdot , t_n \rangle})_{n \in \mathbb{Z}^d}$ is a frame for $L_2[-\pi,\pi]^d$, then $(e^{i \langle \cdot , t_n +\epsilon_n \rangle})_{n \in \mathbb{Z}^d}$ is a frame for $L_2[-\pi,\pi]^d$. These two constants are shown to be asymptotically equivalent for large values of $d$.