Generalized shift-invariant systems and approximately dual frames

Abstract

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Approximately dual pairs of wavelet frames

“…Using standard perturbation results for frames (see, e.g., Theorem 22.1.1 in [6]), it follows that the Riesz basis {D 2 j T k θ} j,k∈Z has lower frame bound A = (1 − η) 2 . Let {ω j,k } j,k∈Z denote the dual Riesz basis associated with {D 2 j T k θ} j,k∈Z .…”

“…In our main result we prove that under a very mild decay condition on the Fourier transform of a function ψ ∈ L 2 (R) that generates a wavelet frame {D a j T kb ψ} j,k∈Z , there exists N ∈ N such that the oversampled wavelet system {D a j T kb/N ψ} j,k∈Z has an approximately dual wavelet frame; most importantly, by choosing N sufficiently large, we can get as close to perfect reconstruction as desired. Approximate duals already appear in a number of contexts in the literature, see the papers [4,2,7,17,18,19]. Also, the parallel case of Gabor frames was considered in the paper [8].…”

Approximately dual frames of vector‐valued nonstationary Gabor frames and reconstruction errors

“…Compared with classical dual frames, approximately dual frames are easier to construct and can be tailored to yield almost perfect reconstruction. Further applications appeared in [10, 27–35] and references therein. In [10], Dörfler and Matusiak considered approximately dual frames of NSG frames for $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$ and provided good approximate reconstruction.…”

“…In [10], Dörfler and Matusiak considered approximately dual frames of NSG frames for $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$ and provided good approximate reconstruction. In [29], Benavente et al investigated approximately dual frames of generalized shift‐invariant systems in $$$ {L}^2\left(\mathrm{\mathbb{R}}\right) $$$ and provided various ways of estimating the deviation from perfect reconstruction. Further, Khedmati and Jakobsen in [35] generalized the results of [29] to generalized translation‐invariant systems on locally compact abelian groups.…”

Approximately Dual p-Approximate Schauder Frames

“…This leads to the notion of approximately dual frames by Christensen and Laugensen [10] which is motivated from the work of Li and Yan [18]. Following the paper [10], there is a plenty of activity on the notion of approximately dual frames and its applications to wavelet frames, Gabor systems, shift-invarinat systems, localized frames, cross Gram matrices [3,4,8,9,[11][12][13]19] etc. This notion is also useful in the study of famous Mexican hat problem [5,6].…”