34 citations · 111 across the 25 of their papers we have counts for
29 papers
The frame set of the first Hermite function
Markus Faulhuber, Philipp Petersen
We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Rome…
Asymptotic safety regions for Gabor frames generated by Hermite functions
Markus Faulhuber, Irina Shafkulovska, Ilya Zlotnikov
The aim of this paper is to establish new regions in the frame sets of Hermite functions . A classical result of Gröchenig and Lyubarskii shows that the Gabor system $\mathcal…
Linear dependence of time-frequency shifts of a Schwartz function
Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven +1
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 Topiwal…
On a fundamental barrier of the Wirtinger criterion for Gabor systems with odd functions
Markus Faulhuber
We show that the Wirtinger criterion cannot be used to investigate the frame set conjecture for the first Hermite function. More generally, for odd functions, it cannot determine r…
The polarization problem for the honeycomb structure
Markus Faulhuber
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 l…
Operator-isomorphism pairs and Zak transform methods for the study of Gabor systems
Markus Faulhuber
We collect and summarize results on the unitary equivalence of Gabor systems by pairs of unitary operators and global isometries. The methods are then used to study Gabor systems w…
Gabor systems with Hermite functions of order n and oversampling greater than n+1 which are not frames
Markus Faulhuber
We show that a sufficient density condition for Gabor systems with Hermite functions over lattices is not sufficient in general. This follows from a result on how zeros of the Zak…
Quantum paving: When sphere packings meet Gabor frames
Markus Faulhuber, Thomas Strohmer
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…
On the frame property of Hermite functions and exploration of their frame sets
Markus Faulhuber, Irina Shafkulovska, Ilya Zlotnikov
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…
A note on energy minimization in dimension 2
Markus Faulhuber, Irina Shafkulovska, Ilia Zlotnikov
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…
Maximal polarization for periodic configurations on the real line
Markus Faulhuber, Stefan Steinerberger
We prove that among all 1-periodic configurations of points on the real line the quantities $$ \min_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2} \quad \text{…
The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators
Markus Faulhuber, Anupam Gumber, Irina Shafkulovska
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. Interest…
Gabor frame bound optimizations
Markus Faulhuber, Irina Shafkulovska
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 squar…
Time-Frequency Analysis (Lecture Notes)
Markus Faulhuber
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…
A variational principle for Gaussian lattice sums
Laurent Bétermin, Markus Faulhuber, Stefan Steinerberger
We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices with fixed density, the minimal value is maximized by the…
Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies
Laurent Bétermin, Markus Faulhuber
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{Glas…
On the optimality of the rock-salt structure among lattices with charge distributions
Laurent Bétermin, Markus Faulhuber, Hans Knüpfer
The goal of this work is to investigate the optimality of the -dimensional rock-salt structure, i.e., the cubic lattice of volume with an alternation o…
The Strohmer and Beaver Conjecture for Gaussian Gabor Systems - A Deep Mathematical Problem (?)
Markus Faulhuber
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…
An Extremal Property of the Hexagonal Lattice
Markus Faulhuber, Stefan Steinerberger
We describe an extremal property of the hexagonal lattice . Let denote the circumcenter of its fundamental triangle (a so-called deep hole) and let …
On the Parity under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes
Markus Faulhuber
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…
An Application of Hypergeometric Functions to Heat Kernels on Rectangular and Hexagonal Tori and a "Weltkonstante" -- Or -- How Ramanujan Split Temperatures
Markus Faulhuber
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 maxi…
Some curious results related to a conjecture of Strohmer and Beaver
Markus Faulhuber
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…
A Short Note on the Frame Set of Odd Functions
Markus Faulhuber
In this work we derive a simple argument which shows that Gabor systems consisting of odd functions of variables and symplectic lattices of density cannot constitute a Ga…
Extremal Determinants of Laplace-Beltrami Operators for Rectangular Tori
Markus Faulhuber
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 thi…