An asymptotic equivalence between two frame perturbation theorems

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 ${\sup }_{n\in {\mathbb{Z}}^d}\Vert {\epsilon }_n{\Vert }_{\infty }<C$, and $(e^{i⟨\cdot ,t_n⟩}{)}_{n\in {\mathbb{Z}}^d}$ is a frame for $L_2[-\pi ,\pi {]}^d$, then $(e^{i⟨\cdot ,t_n+{\epsilon }_n⟩}{)}_{n\in {\mathbb{Z}}^d}$ is a frame for $L_2[-\pi ,\pi {]}^d$. These two constants are shown to be asymptotically equivalent for large values of $d$.