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Markus Faulhuber

Publications and source records attributed to Markus Faulhuber.

At least 19 recordsLinked to original sources
arXiv2026

The frame set of the first Hermite function

We determine the frame set of the first Hermite function, proving a conjecture of Lyubarskii and Nes. We use a characterization of semi-regular Gabor frames due to Gr\"ochenig, Romero, and St\"ockler to transform the problem into a question about the existence of a nonzero Gaussian shift-invariant entire function $F$, with bounded coefficients, whose derivative vanishes on a lattice with spacing $\delta=ab$. By taking the Wronskian of $N$ translates of $F$, we amplify its critical points to zeros of multiplicity at least $N-1$. After rescaling, this Wronskian is again a Gaussian shift-invariant function. A Gaussian zero-density theorem then gives $(N-1)/(N\delta)\leq1$. Varying $N$ over the range allowed by the construction forces every subcritical lattice product $\delta=ab$ for which the system is not a frame to equal $(q-1)/q$ for some integer $q\geq2$.

math.FA↗
arXiv2026

Asymptotic safety regions for Gabor frames generated by Hermite functions

The aim of this paper is to establish new regions in the frame sets of Hermite functions $h_n$. A classical result of Gr\"ochenig and Lyubarskii shows that the Gabor system $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame for $L^2(\mathbb{R})$ whenever the lattice density exceeds $n+1$. We show that, for every $\eta>0$ and all sufficiently large $n$, the same Gabor system forms a frame whenever $ab\leq n^{-\frac{2}{3}-\eta}$. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters $a$ or $b$ is small. Namely, for every $\delta>0$ and $\rho\in(0,\frac{1}{2})$ and all sufficiently large $n$ we prove that if $\min\{a,b\}\leq n^{-\frac{1}{2}-\delta}$ and $ab\leq \frac{1}{2}-\rho$ then $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame.

math.CA↗
arXiv2026

Linear dependence of time-frequency shifts of a Schwartz function

We show that a finite number of time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the so-called HRT conjecture of Heil, Ramanathan, and Topiwala. In particular, we provide an example consisting of 12 time-frequency shifts.

math.FA↗
arXiv2025

The polarization problem for the honeycomb structure

We study the polarization problem in dimension 2 for the honeycomb structure and compare it to the maximal polarization lattice, the hexagonal lattice. As expected, the hexagonal lattice has higher polarization than the honeycomb at all densities.

math.CA↗
arXiv2025

On the frame property of Hermite functions and exploration of their frame sets

We study Gabor frames with Hermite window functions. Gröchenig and Lyubarskii provided a sufficient density condition for their frame sets, which leads to what we call the "safety region". For rectangular lattices and Hermite windows of order 4 and higher, we enlarge this safety region by providing new points on the boundary of this region. For this purpose, we employ the Janssen representation of the frame operator to compare its distance to the identity in the operator norm. The calculations lead to estimates on series involving Laguerre polynomials with Gaussian weight functions.

math.FA↗
arXiv2024

Quantum paving: When sphere packings meet Gabor frames

We introduce the new problems of quantum packing, quantum covering, and quantum paving. These problems arise naturally when considering an algebra of non-commutative operators that is deeply rooted in quantum physics as well as in Gabor analysis. Quantum packing and quantum covering show similarities with energy minimization and the dual problem of polarization. Quantum paving, in turn, aims to simultaneously optimize both quantum packing and quantum covering. Classical sphere packing and covering hint the optimal configurations for our new problems. We present solutions in certain cases, state several conjectures related to quantum paving and discuss some applications.

quant-ph↗
arXiv2024

Maximal polarization for periodic configurations on the real line

We prove that among all 1-periodic configurations $Γ$ of points on the real line $\mathbb{R}$ the quantities $$ \min_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2} \quad \text{and} \quad \max_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $n$ per period is sufficiently large (depending on $α$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$ for large $n$. The first result is shown using Fourier series. The second result follows from work of Cohn and Kumar on universal optimality and holds for all $n$ (independent of $α$).

math.CA↗
arXiv2023

Gabor frame bound optimizations

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 quantities. Also, in some cases optimal lattices do not exist at all and a degenerated system is optimal.

math.FA↗
arXiv2023

A note on energy minimization in dimension 2

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 any scale.

math.CA↗
arXiv2023

Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies

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 result, we show that, among lattices, the hexagonal lattice is the unique maximizer of both families of theta functions. As an immediate consequence, we obtain a new universal optimality result for the hexagonal lattice among two-dimensional alternating charged lattices and lattices shifted by the center of their unit cell.

math.MG↗
arXiv2022

The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators

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 follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence we re-discover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result.

math.CA↗
arXiv2022

Time-Frequency Analysis (Lecture Notes)

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 functional analysis are needed. A course in Fourier analysis may be of advantage, but is not needed. The course contained 4 academic units per week. The appendices and Section 11 were not presented in class.

math.FA↗
arXiv2021

A variational principle for Gaussian lattice sums

We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices $Λ\subset \mathbb{R}^2$ 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. Our inequality resolves a conjecture of Strohmer and Beaver about the operator norm of a certain type of frame in $L^2(\mathbb{R})$. It has implications for minimal energies of ionic crystals studied by Born, the geometry of completely monotone functions and a connection to the elusive Landau constant.

math.CA↗
arXiv2020

Some curious results related to a conjecture of Strohmer and Beaver

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 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. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori.

math.FA↗
arXiv2020

On the optimality of the rock-salt structure among lattices with charge distributions

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 radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the $d$-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials. Numerically, we investigate several particular cases in dimensions $2$, $3$ and $8$. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For $d=2$, we observe a phase transition of the type 'triangular-rhombic-square-rectangular' for the minimizer's shape as the density decreases.

math-ph↗