A Short Note on the Frame Set of Odd Functions

In this work we derive a simple argument which shows that Gabor systems consisting of odd functions of $d$ variables and symplectic lattices of density $2^d$ cannot constitute a Gabor frame. In the 1--dimensional, separable case, this is a special case of a result proved by Lyubarskii and Nes, however, we use a different approach in this work exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.Comment: accepted: Bulletin of the Australian Mathematical Society; 12 pages; Version 3 makes use of symmetric time-frequency shifts. In this case the appearing phase factors are easier to handle. Also, the main result is extended to higher dimensions. [In Version 2 a mistake in the assumptions was corrected. The windows should be chosen from Feichtinger's algebra rather than from the Hilbert space L2.

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

Multi-Window Weaving Frames

In this work we deal with the recently introduced concept of weaving frames. We extend the concept to include multi-window frames and present the first sufficient criteria for a family of multi-window Gabor frames to be woven. We give a Hilbert space norm criterion and a pointwise criterion in phase space. The key ingredient are localization operators in phase space and we give examples of woven multi-window Gabor frames consisting of Hermite functions.Comment: 9 pages, conference paper: SampTA 201

Some curious results related to a conjecture of Strohmer and Beaver

We study results related to a conjecture formulated by Thomas Strohmer and Scott Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace-Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker's limit formula.Comment: 19 pages, 1 figur

Maximal polarization for periodic configurations on the real line

We prove that among all periodic configurations $\Gamma$ of points on the real line $\mathbb{R}$ the quantities \min_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \alpha (x - \gamma)^2} \quad \mbox{and} \quad \max_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \alpha (x - \gamma)^2} are maximized and minimized, respectively, if and only if the points are equispaced points whenever the number of points per period is sufficiently large (depending on $\alpha$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$. The first result is shown using Fourier series, the second follows from work of Cohn-Kumar on universal optimality