Markus Faulhuber

University of Vienna | UniWien · Fakultät für Mathematik

We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices $\Lambda \subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be interpreted as the dual of a 1988 result of Montgomery who proved that the hexagonal lattice minimizes the maximal values....

We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This...

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 i...

Proving the universal optimality of the hexagonal lattice is one of the big open challenges of nowadays mathematics. We show that the hexagonal lattice outperforms certain "natural" classes of periodic configurations. Also, we rule out the option that the canonical non-lattice rival -- the honeycomb -- has lower energy than the hexagonal lattice at...

We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the frame bounds and the condition number are rectangular lattices which are different for the respective quantiti...

We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H. Montgomery, Minimal theta functions. Glasgow Mathematical Journal, 30 (1988), 75–85]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery’s result, we show...

We study sharp frame bounds of Gabor systems over rectangular lattices for different windows. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the frame bounds and the condition number are different. Also, in some cases optimal lattices do not exist at all and a degenerated system is op...

These lecture notes accompanied the course Time-Frequency Analysis given at the Faculty of Mathematics of the University of Vienna in the summer term 2021. The material is suitable for an advanced undergraduate course in mathematics or a mathematics class for PhD students. Besides standard linear algebra and calculus only some basics from functiona...

In this work we study the determinant of the Laplace–Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments. Refined logarithmic convexity and concavity results...

In this work we investigate the heat kernel of the Laplace-Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss' hypergeometric function 2 F 1 and the elliptic modulus. In order to be able to do this, we...

The goal of this paper is to investigate the optimality of the [Formula: see text]-dimensional rock-salt structure, i.e. the cubic lattice [Formula: see text] of volume [Formula: see text] with an alternation of charges [Formula: see text] at lattice points, among periodic distributions of charges and lattice structures. We assume that the charges...

We study results related to a conjecture formulated by Strohmer and 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 a...

We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H.~Montgomery. Minimal theta functions. \textit{Glasgow Mathematical Journal}, 30(1):75--85, 1988]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery's resu...

The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$ at lattice points, among periodic distribution of charges and lattice structures. We assume that the charges are interacting through two types of radial...

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a “phase space frame” associated with a Weyl- Heisen...

In this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes,...

We describe an extremal property of the hexagonal lattice Λ⊂R2. Let p denote the circumcenter of its fundamental triangle (a so-called deep hole) and let Ar denote the set of lattice points that are at distance r from pAr=λ∈Λ:‖λ-p‖=r.If Γ is a small perturbation of Λ in the space of lattices with fixed density and Cr denotes the set of points in Ar...

In this article we are going to discuss the conjecture of Strohmer and Beaver for Gaussian Gabor systems. It asks for an optimal sampling pattern in the time-frequency plane, where optimality is measured in terms of the condition number of the frame operator. From a heuristic point of view, it seems obvious that a hexagonal (sometimes called triang...

We describe an extremal property of the hexagonal lattice $\Lambda \subset \mathbb{R}^2$. Let $p$ denote the circumcenter of its fundamental triangle (a so-called deep hole) and let $A_r$ denote the set of lattice points that are at distance $r$ from $p$ \begin{equation} A_r = \left\{ \lambda \in \Lambda: \| \lambda - p \| = r\right\}. \end{equatio...

In this work we show that if the frame property of a Gabor frame with window in Feichtinger's algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes,...

In this work we investigate the heat kernel of the Laplace-Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss' hypergeometric function 2F1 and the elliptic modulus. In order to be able to do this, we employ...

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...

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 i...

In this work we study the determinant of the Laplace-Beltrami operator on rectangular tori of unit area. We will see that the square torus gives the extremal determinant within this class of tori. The result is established by studying properties of the Dedekind eta function for special arguments and refined logarithmic convexity and concavity resul...

We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or equivalently the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.

In this work we study and collect symmetry properties of the classical Jacobi theta functions. These properties concern the logarithmic derivatives of Jacobi's theta functions on a logarithmic scale.

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 operat...

We study sharp frame bounds of Gabor frames with the standard Gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all rectangular lattices. This proves a conjecture of Floch, Alard & Berrou (as reformulated by Strohmer & Beaver). The proof is based on refined log-convexity/concavity estimates f...

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...

presented at ”Winter School on Advances in Mathematics of Signal Processing”

presented at the ”February Fourier Talks 2015”

We want to investigate the problem whether global aspects of a Gabor frame, i.e. the frame condition can be predicted by local properties, i.e. the geometry of the underlying lattice. Using the L^2 normalised standard Gaussian as window function, experimental results suggest a strong connection between packing problems and Gabor frames. Hamiltonian...

In the underlying work we will point out conjectured connections between Gabor frames and geometric properties of lattices. The concept of Gabor frame is a certain kind of time–frequency representation method and as such underlying the uncertainty principles. This means that the product of a signal’s length and its bandwidth cannot be arbitrarily s...