We study spanning properties of a family functions translated along simple model sets. characterize tight frame and dual generators for such irregular translates we apply the results to Gabor systems. use connection between sets almost periodic rely strongly on Poisson summations formula introduce so-called bracket product, which then plays crucial role in our approach. As corollary main obtain...

Single-frame multichannel blind deconvolution is formulated by applying a bank of Gabor filters to a blurred image. The key observation is that spatially oriented Gabor filters produce sparse images and that a multichannel version of the observed image can be represented as a product of an unknown nonnegative sparse mixing vector and an unknown nonnegative source image. Therefore a blind-deconv...

Let {xn} be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {xn} which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {xn} can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is ...

We characterize L norms of functions on R for 1 < p < ∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L functions converge to the functions almost everywhere and in L for 1 < p < ∞. In L we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L in a certain Cesàro sense. Consequ...

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces ...

Approximately dual frames are studied in the Hilbert space setting. Approximate duals are easier to construct than classical dual frames, and can be tailored to yield almost perfect reconstruction. Bounds on the deviation from perfect reconstruction are obtained for approximately dual frames constructed via perturbation theory. An alternative bound is derived for the rich class of Gabor frames,...

We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to yield almost perfect reconstruction. For approximately dual frames constructed via perturbation theory, we provide a bound on the deviation from perfect recon...