Nonstationary Gabor frames – approximately dual frames and reconstruction errors. (English) Zbl 1345.42031
Summary: Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper, it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual frames. In particular, we give constructive examples for so-called almost painless nonstationary frames, that is, frames that are closely related to nonstationary frames with compactly supported windows. The theoretical results are illustrated by concrete computational and numerical examples.
MSC:
| 42C15 | General harmonic expansions, frames |
| 41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
| 42C30 | Completeness of sets of functions in nontrigonometric harmonic analysis |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 65T60 | Numerical methods for wavelets |
Keywords:
adaptive representations; nonorthogonal expansions; irregular Gabor frames; reconstruction; approximately dual frameReferences:
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