This chapter illustrates the use of Gabor frame analysis to derive results on the spectral properties of integral and pseudodifferential operators. In particular, we obtain a sufficient condition on the kernel of an integral operator or the symbol of a pseudodifferential operator which implies that the operator is trace-class. This result significantly improves a sufficient condition due to Dau...

Using a variant of the Sobolev Embedding Theorem, we prove an uncertainty principle related to Gabor systems that generalizes the Balian-Low Theorem. Namely, if f ∈ H(R) and f̂ ∈ H /2(R) with 1 < p < ∞, 1 p + 1 p′ = 1, then the Gabor system G(f, 1, 1) is not a frame for L(R). In the p = 1 case, we obtain a generalization of the result in [BCPS].

We show that multi-window Gabor frames with windows in the Wiener algebra W (L∞, `) are Banach frames for all Wiener amalgam spaces. As a by-product of our results we prove the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra also belongs to this space. Our proofs are mostly based on recent noncommutative versions of Wiener’s 1/f lemma.

To overcome the limitation induced by the fixed timefrequency resolution over the whole time-frequency plane of Gabor frames, we propose a simple extension of the Gabor theory leading to the construction of frames with timefrequency resolution evolving over time or frequency. We describe the construction of such frames and give the explicit formulation of the canonical dual frame for some condi...

In this paper, we generalized Daubechies'[2] results on 1D wavelets to 2D, and elucidated the conditions in which a family of Gabor wavelets will provide a complete representation of an image. We found that the phase space (scale, orientation, x, y) sampling density of the striate cortical simple cells is sufficiently dense to form a tight wavelet frame. This tight frame approximates the contin...

The main purpose of the paper is to give a characterization all compactly supported dual windows Gabor frame. As an application, we consider iterative procedure for approximation canonical window via on every step. In particular, allows have from certain modulation spaces or Schwartz space.

In this paper we present an attractive computation technique for approximating the continuous-time dual window for the Gabor signal expansion on a non-separable time-frequency lattice instead of on the traditional rectangular lattice. This non-separable lattice is described by means of a lattice generator matrix. The computation technique is based on a generalized Wexler-Raz identity and avoids...

We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion‎. ‎In particular, we prove a version of Plancherel-Po ́lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity‎, ‎exactness and deficienc...