14 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…
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…
Stable Phase Retrieval for Spans of Independent Random Variables
Pedro Abdalla, Jaume de Dios Pont, João P. G. Ramos +1
We prove that, after normalization, stable phase retrieval holds over the -spans of independent real-valued centered random variables if and only if all but possibly one…
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…
Maximal inequalities, frames and greedy algorithms
Pablo Berná, Daniel Freeman, Timur Oikhberg +1
The aim of this article is to use Banach lattice techniques to study coordinate systems in function spaces. We begin by proving that the greedy algorithm of a basis is order conver…
The Cahill-Casazza-Daubechies problem on Hölder stable phase retrieval
Daniel Freeman, Mitchell A. Taylor
Phase retrieval using a frame for a finite-dimensional Hilbert space is known to always be Lipschitz stable. However, phase retrieval using a frame or a continuous frame for an inf…
On sharp stable recovery from clipped and folded measurements
Pedro Abdalla, Daniel Freeman, João P. G. Ramos +1
We investigate the stability of vector recovery from random linear measurements which have been either clipped or folded. This is motivated by applications where measurement device…