arXiv subjects

Torgeir Keun Lysen

Publications and source records attributed to Torgeir Keun Lysen.

3 recordsLinked to original sources

arXiv2026

Discrete Pauli pairs

We determine sharp density thresholds for when discrete Pauli pairs with Gaussian decay must be classical Pauli pairs. More precisely, we identify when equality of the sampled moduli of two functions and their Fourier transforms forces equality of the global moduli in time and frequency, thereby answering a question of Ramos and Sousa. We also determine the sharp threshold for when discrete Pauli pairs must be weak Pauli pairs, where either the time-side or the frequency-side moduli agree identically. These can both be seen as phaseless versions of the results of Kulikov, Nazarov, and Sodin on Fourier uniqueness pairs.

math.CA↗

arXiv2026

A discrete Hardy uncertainty principle

We show that knowing the decay of a function $f$ on a discrete set $Λ\subset\mathbb{R}$ and the decay of its Fourier transform $\hat{f}$ on a discrete set $M\subset\mathbb{R}$ is enough to determine the global decay of $f$ and $\hat{f}$, provided that $(Λ,M)$ is a supercritical pair in the sense of Kulikov, Nazarov, and Sodin. This decay transfer result leads to a discrete generalization of Morgan's uncertainty principle: it is enough to require $|f(λ)|\lesssim e^{-\frac{2}{p}Aπ|λ|^p}$ for all $λ\inΛ$ and $|\hat{f}(μ)|\lesssim e^{-\frac{2}{q}Aπ|μ|^q}$ for all $μ\in M$, where $(p,q)$ are Hölder conjugates, $A>|\cos(\frac{rπ}{2})|^\frac{1}{r}$, and $r:=\min\{p,q\}$. For $A=1$ and $p,q=2$, we also show that any such function must be a scaled Gaussian. This yields a discrete version of Hardy's uncertainty principle and resolves two questions posed by Ramos and Sousa.

math.CA↗

arXiv2025

Critical and asymmetric Fourier uniqueness pairs

Motivated by the recent work of Kulikov, Nazarov, and Sodin, we construct sufficient conditions for discrete subsets of $\mathbb{R}$, which lie between the supercritical and subcritical cases, to constitute Fourier uniqueness pairs. This family of critical uniqueness pairs includes pairs that are strongly asymmetric, stretching beyond those associated with zeros of zeta and L-functions, discovered by Bondarenko, Radchenko, and Seip, and getting arbitrarily close to the classical Shannon--Whittaker uniqueness pair. We also identify a somewhat more restrictive family of strongly asymmetric uniqueness pairs that yield frames and hence Fourier interpolation.

math.CA↗