## Pairs of dual Gabor frame generators with compact support and desired frequency localization.(English)Zbl 1106.42030

Denote $$(T_a f)(x)=f(x-a)$$ and $$(E_b f)(x)=e^{2\pi i bx} f(x)$$. Let $$g$$ be a real-valued bounded function in $$L^2(\mathbb{R})$$ such that $$\sum_{k\in \mathbb{Z}} g(x-k)=1$$ and $$g$$ is supported inside $$[0,N]$$ for some integer $$N$$. This paper shows in Theorems 2.2 and 2.6 that for any $$b\in (0, \frac{1}{2N-1})$$, $$\{E_{mb} T_n g\}_{m,n\in \mathbb{Z}}$$ and $$\{E_{mb} T_n h\}_{m,n\in \mathbb{Z}}$$ form a pair of dual Gabor dual frames for $$L^2(\mathbb{R})$$, where $$h(x)=bg(x)+2b \sum_{n=1}^{N-1} g(x+n)$$.
Reviewer: Bin Han (Edmonton)

### MSC:

 42C99 Nontrigonometric harmonic analysis 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

### Keywords:

Gabor frames; dual frames; dual generators; wavelet frames
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### References:

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