Aldroubi has shown how one can construct any frame {gi}i=1 starting with one frame {fi}i=1,using a bounded operator U on l2(N). We study the overcompleteness of the frames in terms of properties of U . We also discuss perturbation of frames in the sense that two frames are “close” if a certain operator is compact. In this way we obtain an equivalence relation with the property that members of t...

In this paper the relativistic quantum mechanics is considered in the framework of the nonstandard synchronization scheme for clocks. Such a synchronization preserves Poincaré covariance but (at least formally) distinguishes an inertial frame. This enables to avoid the problem of a noncausal transmision of information related to breaking of the Bell's inequalities in QM. Our analysis has been f...

The Laplacian pyramid (LP) provides a frame expansion. Thus, there exist infinitely many synthesis operators which achieve perfect reconstruction in the absence of quantization. However, if the subbands are quantized in the open-loop mode then the dual frame synthesis operator, which is the pseudo-inverse of the analysis operator, minimizes the mean squared error (MSE) in the reconstruction. No...

We use a generalization of Wiener’s 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space W (L∞, ` )(R), the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the generator of the canonical dual belongs also to W (L∞, ` )(R).

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame is a frame-like collection of subspaces in a Hilbert space, thereby generalizing the concept of a frame for signal representation. In this paper, we study the existence and construction of fusion frames. We first present a complete characterization...

The concept of g-frame is a natural extension the frame. This article mainly discusses relationship between some special bounded linear operators and g-frames, characterizes properties g-frames. In addition, according to operator spectrum theory, eigenvalues are introduced into new expression best frame boundary given.

Given a real and invertible d×d matrix C, we define for k ∈ Zd a generalized translation operator TCk acting on f ∈ L 2(Rd) by (TCkf)(x) = f(x − Ck), x ∈ R . A generalized shift-invariant system is a system of the type {TCjkφj}j∈J,k∈Zd , where {Cj}j∈J is a countable collection of real invertible d×d matrices, and {φj}j∈J ⊂ L 2(Rd). Generalized shift-invariant systems contain the classical wavel...

Fusion frames, and, more generally, operator-valued frame sequences are generalizations of classical frames, which are today a standard notion when redundant, yet stable sequences are required. However, the question of stability of duals with respect to perturbations has not been satisfactorily answered. In this paper, we quantitatively measure this stability by considering the associated devia...