Weyl-Heisenberg frames for subspaces of 𝐿²(𝑅)

Casazza, Peter; Christensen, Ole

doi:10.1090/S0002-9939-00-05731-2

Weyl-Heisenberg frames for subspaces of $L^2(R)$
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by Peter G. Casazza and Ole Christensen PDF

Proc. Amer. Math. Soc. 129 (2001), 145-154 Request permission

Abstract:

A Weyl-Heisenberg frame \[ \{E_{mb}T_{na}g \}_{m,n \in Z} = \{ e^{2 \pi imb ( \cdot ) } g( \cdot - na) \}_{m,n \in Z}\] for $L^2(R)$ allows every function $f \in L^2(R)$ to be written as an infinite linear combination of translated and modulated versions of the fixed function $g \in L^2(R)$. In the present paper we find sufficient conditions for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $\overline {span}\{E_{mb}T_{na}g \}_{m,n \in Z}$, which, in general, might just be a subspace of $L^2(R)$. Even our condition for $\{E_{mb}T_{na}g\}_{m, n \in Z}$ to be a frame for $L^2(R)$ is significantly weaker than the previous known conditions. The results also shed new light on the classical results concerning frames for $L^2(R)$, showing for instance that the condition $G(x):= \sum _{n \in Z} | g(x-na)|^2 >A>0$ is not necessary for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $\overline {span} \{E_{mb}T_{na}g \}_{m,n \in Z}$. Our work is inspired by a recent paper by Benedetto and Li, where the relationship between the zero-set of the function $G$ and frame properties of the set of functions $\{g( \cdot - n) \}_{n \in Z}$ is analyzed.

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