The effect of perturbations of frames on their alternate and approximately dual frames

Abstract

Approximately dual frames as a generalization of duality notion in Hilbert spaces have applications in Gabor systems, wavelets, coorbit theory, and sensor modeling. In recent years, the computing of the associated deviations of the canonical and alternate dual frames from the original ones has been considered by functional analysts. In this paper, we consider the quantitative measurement of the associated deviations of the alternate and approximately dual frames from the original ones. More precisely, it is proved that if the sequence Ψ is sufficiently close to the frame Φ, then for given approximately dual frame Φ ad of Φ, the approximately dual frame Ψ ad of Ψ can be found which is close to Φ ad , and particularly, we estimate their deviation in terms of the distance of the operators  1 ∶= T Φ U Φ ad and  2 ∶= T Ψ U Ψ ad and their approximation rates, where T X and U X denote the synthesis and analysis operators of the frame X, respectively. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literatures on this topic.

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.…”

“…Further, Khedmati and Jakobsen in [35] generalized the results of [29] to generalized translation-invariant systems on locally compact abelian groups. Recently, approximately dual pairs of wavelet frames was discussed in [28], a quantitative measurement of the deviations of the approximately dual frames from the original ones was given in [34], and approximate oblique dual frames were investigated in [31].…”