Characterization and computation of canonical tight windows for Gabor frames

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples

Gabor Frame Sets of Invariance - A Hamiltonian Approach to Gabor Frame Deformations

In this work we study families of pairs of window functions and lattices which lead to Gabor frames which all possess the same frame bounds. To be more precise, for every generalized Gaussian $g$, we will construct an uncountable family of lattices $\lbrace \Lambda_\tau \rbrace$ such that each pairing of $g$ with some $\Lambda_\tau$ yields a Gabor frame, and all pairings yield the same frame bounds. On the other hand, for each lattice we will find a countable family of generalized Gaussians $\lbrace g_i \rbrace$ such that each pairing leaves the frame bounds invariant. Therefore, we are tempted to speak about "Gabor Frame Sets of Invariance".Comment: To appear in "Journal of Pseudo-Differential Operators and Applications

Extreme Singular Values of Random Time-Frequency Structured Matrices

In this paper, we investigate extreme singular values of the analysis matrix of a Gabor frame $(g, \Lambda)$ with a random window $g$. Columns of such matrices are time and frequency shifts of $g$, and $\Lambda\subset \mathbb{Z}_M\times\mathbb{Z}_M$ is the set of time-frequency shift indices. Our aim is to obtain bounds on the singular values of such random time-frequency structured matrices for various choices of the frame set $\Lambda$, and to investigate their dependence on the structure of $\Lambda$, as well as on its cardinality. We also compare the results obtained for Gabor frame analysis matrices with the respective results for matrices with independent identically distributed entries.Comment: 21 pages, 3 figure

Gabor Frame Decomposition of Evolution Operators and Applications

We compute the Gabor matrix for Schr\"odinger-type evolution operators. Precisely, we analyze the Heat Equation, already presented in \cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix which leads to better numerical evaluations. Then, using asymptotic integration techniques, we obtain an upper bound for the Gabor matrix in one-dimension for the generalized Heat Equation, new in the literature. Using Maple software, we show numeric representations of the coefficients' decay. Finally, we show the super-exponential decay of the coefficients of the Gabor matrix for the Harmonic Repulsor, together with some numerical evaluations. This work is the natural prosecution of the ideas presented in \cite{2012arXiv1209.0945C} and \cite{MR2502369}.Comment: 29 pages, 7 figure

The abcabc-problem for Gabor systems

A Gabor system generated by a window function $\phi$ and a rectangular lattice $a \Z\times \Z/b$ is given by $${\mathcal G}(\phi, a \Z\times \Z/b):=\{e^{-2\pi i n t/b} \phi(t- m a):\ (m, n)\in \Z\times \Z\}.$$ One of fundamental problems in Gabor analysis is to identify window functions $\phi$ and time-frequency shift lattices $a \Z\times \Z/b$ such that the corresponding Gabor system ${\mathcal G}(\phi, a \Z\times \Z/b)$ is a Gabor frame for $L^2(\R)$, the space of all square-integrable functions on the real line $\R$. In this paper, we provide a full classification of triples $(a,b,c)$ for which the Gabor system ${\mathcal G}(\chi_I, a \Z\times \Z/b)$ generated by the ideal window function $\chi_I$ on an interval $I$ of length $c$ is a Gabor frame for $L^2(\R)$. For the classification of such triples $(a, b, c)$ (i.e., the $abc$-problem for Gabor systems), we introduce maximal invariant sets of some piecewise linear transformations and establish the equivalence between Gabor frame property and triviality of maximal invariant sets. We then study dynamic system associated with the piecewise linear transformations and explore various properties of their maximal invariant sets. By performing holes-removal surgery for maximal invariant sets to shrink and augmentation operation for a line with marks to expand, we finally parameterize those triples $(a, b, c)$ for which maximal invariant sets are trivial. The novel techniques involving non-ergodicity of dynamical systems associated with some novel non-contractive and non-measure-preserving transformations lead to our arduous answer to the $abc$-problem for Gabor systems

Time-Frequency Shift Invariance of Gabor Spaces with an S0S_0-Generator

We consider non-complete Gabor frame sequences generated by an $S_0$-function and a lattice $\Lambda$ and prove that there is $m \in \mathbb{N}$ such that all time-frequency shifts leaving the corresponding Gabor space invariant have their parameters in $\tfrac{1}{m}\Lambda$. We also investigate time-frequency shift invariance under duality aspects