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Published in:
2012 | OriginalPaper | Chapter
An Asymptotic Equivalence Between Two Frame Perturbation Theorems
Author : B. A. Bailey
Published in: Approximation Theory XIII: San Antonio 2010
Publisher: Springer New York
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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
.