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

Christensen, Ole

Appl. Comput. Harmon. Anal. 20, No. 3, 403-410 (2006).

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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