30 citations · 127 across the 22 of their papers we have counts for
25 papers
Structured coordinate systems in
Daniel Freeman, Lukas Liehr, Mitchell A. Taylor
We construct Schauder bases of with significant structural constraints on their elements. First, we construct non-negative Schauder bases of for…
Phase retrieval from a uniformly discrete point set
Jaume de Dios Pont, Josef Greilhuber, Lukas Liehr +1
We prove that for every window of the form , where is a polynomial, there exists a uniformly discrete set of points $\mathcal{S} \subset \mathbb R^…
Counterexamples to Grad's conjecture
Javier Gómez-Serrano, Lukas Liehr, Mitchell A. Taylor
For each sufficiently large , we construct smooth solutions of the magnetohydrostatic equations on embedded solid tori whose regular pressure levels are nested tori, whose magne…
Energy minimization for eight points on the sphere
Liudmyla Kryvonos, Lukas Liehr, Mitchell A. Taylor
We study the energy minimization problem for eight points on the unit sphere. For the logarithmic and Coulomb energies, we show that the unique global minimizer up to congruence is…
Universal completeness of exponentials
Susanna Bertolini, Enric Florit-Simon, Lukas Liehr +1
We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets , of uniform density one, such…
Derivatives of Theta Functions and a Problem of Lyubarskii and Nes
Lukas Liehr, Irina Shafkulovska, Mitchell A. Taylor
We characterize all lattices of rational density for which the Gabor system generated by the first Hermite function along is a frame for $L^2(\mathbb{R}…
Grothendieck's theorem for Bessel sequences
Lukas Liehr, Mitchell A. Taylor, Peiyang Yu
We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence with Bessel bound in a Hilbert spac…
Banach lattices and phase retrieval: A case study for the use of AI in mathematics
Jaume de Dios Pont, Lukas Liehr, David Muñoz-Lahoz +2
The ability of large language models to assist professional mathematicians has been progressing rapidly. Earlier this year, a group of researchers in Banach lattice theory and phas…
Gabor Frames of Totally Positive Functions: A Complete Characterization
Jaume de Dios Pont, Karlheinz Gröchenig, Lukas Liehr +2
We prove that the set of time-frequency shifts with a continuous, integrable totally positive function and lattice parameters $α…
Cantor measures with odd base do not admit Fourier frames
Jaume de Dios Pont, Lukas Liehr, Mitchell A. Taylor
We prove that the Cantor measure with base does not admit a Fourier frame whenever is an odd integer. In particular, this answers a question of Strichartz on the existe…
On the existence problem of regular Gabor frames
Jaume de Dios Pont, Lukas Liehr, Mitchell A. Taylor
For every dimension , we establish explicit criteria on lattices with density such that no function with a continuous Zak transform gen…
Compactly supported Gabor orthonormal bases
Lukas Liehr
We characterize all lattices and all compactly supported functions for which the Gabor system $\left \{ e^{2πi s x} g(x-t) : (t,s) \…
Cheeger's isoperimetric problem for Gaussian mixtures
Lukas Liehr
In any dimension , we determine the Cheeger constant and the Cheeger sets of the Gaussian mixture , where , , an…
Nonlinear determination and phase retrieval under unimodular constraints
Lukas Liehr, Tomasz Szczepanski
We study nonlinear determination problems in Hilbert spaces in which inner products are observed up to prescribed rotations in the complex plane. Given a Hilbert space and a su…
Translation-based completeness on compact intervals
Lukas Liehr
Given a compact interval , and a function that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot -…
Arithmetic progressions and holomorphic phase retrieval
Lukas Liehr
We study the determination of a holomorphic function from its absolute value. Given a parameter , we derive the following characterization of uniqueness in terms o…
Phase retrieval in Fock space and perturbation of Liouville sets
Philipp Grohs, Lukas Liehr, Martin Rathmair
We study the determination of functions in Fock space from samples of their absolute value, known as the phase retrieval problem in Fock space. An important finding in this researc…
From completeness of discrete translates to phaseless sampling of the short-time Fourier transform
Philipp Grohs, Lukas Liehr, Irina Shafkulovska
We study the uniqueness problem in short-time Fourier transform phase retrieval by exploring a connection to the completeness problem of discrete translates. Specifically, we prove…
Phaseless sampling on square-root lattices
Philipp Grohs, Lukas Liehr
Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval…
Multi-window STFT phase retrieval: lattice uniqueness
Philipp Grohs, Lukas Liehr, Martin Rathmair
Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function from its spectrogram, i.e., the magnitudes of its short-time Fourier transform $V…
Non-uniqueness theory in sampled STFT phase retrieval
Philipp Grohs, Lukas Liehr
The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important application…
On foundational discretization barriers in STFT phase retrieval
Philipp Grohs, Lukas Liehr
We prove that there exists no window function and no lattice such that every is determined up to…
Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
Philipp Grohs, Lukas Liehr
We study the phase reconstruction of signals belonging to complex Gaussian shift-invariant spaces from spectrogram measurements where $\mathc…
Injectivity of Gabor phase retrieval from lattice measurements
Philipp Grohs, Lukas Liehr
We establish novel uniqueness results for the Gabor phase retrieval problem: if denotes the Gabor transform then every $f \in…