Gabor Frame Operator Research Articles

Discrete Gabor structures and optimal representations

The idea of Gabor's (1946) signal expansion is to represent a signal in terms of a discrete set of time-shifted and frequency modulated signals that are localized in the time-frequency (or phase) space. We present detailed descriptions of the block and banded structures for the Gabor matrices. Based on the explicit descriptions of the sparsity of such matrices, we can establish the sparse form of the Gabor matrix and obtain the dual Gabor atom (mother wavelet), the inverse of the Gabor frame operator, and carry out the discrete finite Gabor transform in a very efficient way. Some explicit sufficient and also necessary conditions are derived for a Gabor atom g to generate a Gabor frame with respect to a TF-lattice (a, b). >

Lattice size estimates for Gabor decompositions

It is shown that the Gabor frame operator for frames with lattice such thatab=1/N for some integerN≥1 is continuous on the modulation spaces ([9], [20]) whenever the basic functiong is a Schwartz function and (g, a, b) generates a frame forL2 (R). From this a sufficient condition on the lattice is given which guarantees that one has a Banach frame for the modulation spaces of Feichtinger.

Continuity properties of the Gabor frame operator

For a Gabor frame with small lattice, it is shown that the frame operator is continuous and invertible on many Banach spaces defined by smoothness and decay properties. As a consequence, it is shown that there exist Gabor-type frame/dual frame pairs and tight frames with good decay and smoothness properties. Also, it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly (e.g., in Lp and Sobolev norms). Also, explicit estimates on lattice sizes for which such results hold can be obtained.

Multi-window Gabor frames in amalgam spaces

We show that multi-window Gabor frames with windows in the Wiener algebra $W(L^{\infty}, \ell^{1})$ are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wiener's $1/f$ lemma.