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 Sun and Zhou (J. Math. Anal. Appl. 235:159–167, 1999)), each give a constant such that if ${\sup }_{n\in {\mathbb{Z}}^{}}\|{\epsilon {}_{n}\|}_{\infty } < C$, and (e^{i⟨ ⋅,t} _n ^⟩) _n∈ℤ ^d is a frame for L ₂[−π,π]^d, then (e^{i⟨ ⋅,t} _n +ε _n ^⟩) _n∈ℤ ^d is a frame for L ₂[−π,π]^d. These two constants are shown to be asymptotically equivalent for large values of d.