We derive frame estimates for vector-valued Gabor systems with window functions belonging to Schwartz space. The main result provides frame bound estimates for windows composed of Hermite functions. The proof is based on a recently established sampling theorem for the simply connected Heisenberg group, which is translated to a family of frame estimates via a direct integral decomposition.

The Density Theorem for Gabor Frames is a fundamental result in timefrequency analysis. Beginning with Baggett’s proof that a rectangular lattice Gabor system {eg(t−αk)}n,k∈Z must be incomplete in L (R) whenever αβ > 1, the necessary conditions for a Gabor system to be complete, a frame, a Riesz basis, or a Riesz sequence have been extended to arbitrary lattices and beyond. The first partial pr...

We show that Hilbert–Schmidt operators can be used to define frame-like structures for $L^2(\mathbb{R}^d )$ over lattices in $\mathbb{R}^{2d} $ include multi-window Gabor frames as a special case. These are called g-frames

Signal analysis with classical Gabor frames leads to a fixed time-frequency resolution over the whole time-frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time-frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames a...

Regular Gabor frames for L2(Rd) are obtained by applying time-frequency shifts from a lattice in Λ Rd× R̂d to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some g∈ S0(R). There is always a canonical dual frame, generated by the dual Gabor atom g̃. The paper promotes a numerical approach f...

A class of non-canonical duals for the multiwindow Gabor scheme, incorporating both rational and integer oversampling of the Gaborian combined time-frequency space, is considered. Using properties of Gabor frame matrices, block Discrete Fourier transforms, and results from number theory, we establish existence conditions for the noncanonical duals for both integer and rational oversampling rate...

A class of non-canonical duals for the multiwindow Gabor scheme, incorporating both rational and integer oversampling of the Gaborian combined time-frequency space, is considered. Using properties of Gabor frame matrices, block Discrete Fourier transforms, and results from number theory, we establish existence conditions for the noncanonical duals for both integer and rational oversampling rate...

Many results and problems in Fourier and Gabor analysis are formulated in the continuous variable case, i.e., for functions on R. In contrast, a suitable setting for practical computations is the finite case, dealing with vectors of finite length. We establish fundamental results for the approximation of the continuous case by finite models, namely, the approximation of the Fourier transform an...

Visual Voice Activity Detection (VVAD) refers to the detection of speech from a video sequence by means of visual cues. VVAD provides a useful addition to auditory voice activity detection, in particular in cases involving multiple speakers or background noise. This paper focusses explicitly on the measurement of facial movements at different speeds to determine which rates of movement contribu...