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The paper considers the basis and frame properties of the system of weighted exponentials ${\mathcal{E}}(g,\mathbb{Z}\backslash F) = \{e^{2\pi i n x} g(x)\}_{n\in\mathbb{Z}\backslash F}$ in $L^{2}({\mathbb{T}})$ , where $g \in L^{2}({\mathbb{T}}) \backslash\{0\}$ and F⊂ℤ. It is shown that many of the frame properties of ${\mathcal {E}}(g,\mathbb{Z}\backslash F)$ are affected by the cardinalities of F and the behavior of the zeros of g.
Acta Applicandae Mathematicae – Springer Journals
Published: Dec 16, 2011
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