Frame set for Gabor systems with Haar window. (English) Zbl 1537.42053
Summary: We describe the full structure of the frame set for the Gabor system \(\mathcal{G}(g; \alpha, \beta) := \{e^{-2 \pi im \beta \cdot} g(\cdot - n \alpha) : m, n \in \mathbb{Z}\}\) with the window being the Haar function \(g = - \chi_{[-1/2, 0)} + \chi_{[0, 1/2)}\). This is the first compactly supported window function for which the frame set is represented explicitly.
The strategy of this paper is to introduce the piecewise linear transformation \(\mathcal{M}\) on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system \(\mathcal{G}(g; \alpha, \beta)\) to be a frame, i.e., the symmetric invariant set of the transformation \(\mathcal{M}\) is empty.
The strategy of this paper is to introduce the piecewise linear transformation \(\mathcal{M}\) on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system \(\mathcal{G}(g; \alpha, \beta)\) to be a frame, i.e., the symmetric invariant set of the transformation \(\mathcal{M}\) is empty.
MSC:
| 42C15 | General harmonic expansions, frames |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 28D05 | Measure-preserving transformations |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 94A20 | Sampling theory in information and communication theory |
Keywords:
Gabor frame; Haar function; piecewise linear transformation; maximal invariant set; symmetric maximal invariant setReferences:
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