Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.27846

Derivatives of Theta Functions and a Problem of Lyubarskii and Nes

Lukas Liehr·Irina Shafkulovska·Mitchell A. Taylor
View original source ↗Read source document ↗Export metadata ↓

Abstract

We characterize all lattices $Λ\subset \mathbb{R}^2$ of rational density for which the Gabor system generated by the first Hermite function along $Λ$ is a frame for $L^2(\mathbb{R})$. We prove that these are precisely the lattices whose density satisfies $D(Λ) = \frac{q}{p} $ where $p,q \in \mathbb{N}$ are coprime with $q \geq p+2$, thereby confirming a conjecture of Lyubarskii and Nes. A formalization of our main result in Lean $4$ is also included.

Explore related subjects

math.CAmath.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lukas Liehr, Irina Shafkulovska, Mitchell A. Taylor. 2026-08-27. Derivatives of Theta Functions and a Problem of Lyubarskii and Nes. https://arxiv.org/abs/2609.27846

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A discussion of three arguments related to Fefferman's Fourier extension theorem in the plane

The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sjölin, with simplifications given by other authors later on, in particular by L. Hörmander and T. Tao. We discuss yet two more arguments for this classical theorem on the parabola. The first argument uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second argument whose point of departure is the bilinear characterization of Tao, Vargas and Vega, and relies on the bilinear interplay with the classical wave packet constructions and discrete characterizations with an induction on scales. However, each of the above two arguments rely on some form of Fefferman's convolution decoupling and the special nature of the critical planar index 4 as a positive even integer. On the other hand, our third argument essentially avoids both of these obstacles by using smooth Alpert projections with wave packets, discrete bilinear characterizations, and the discrete Fourier transform of the coefficient sequences associated with the projections.

math.CA↗
arXiv2026

Hypergeometric Mixed-Type Multiple Orthogonal Polynomials

Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one $q\times p$ matrices of weights, with arbitrary $p$ and $q$: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at $(x,1)$. For the mixed beta--Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer $G$-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with $p$ subdiagonals and $q$ superdiagonals. All recurrence coefficients are given by finite Gamma--Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss--Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.

math.CA↗
arXiv2026

Sharp power-mean comparisons for Gauss hypergeometric functions

We determine the sharp weighted power-mean comparisons for the family $\{H_a(r)\}_{r\in (0,1)}$ with the logarithmic interpretation at $a=0$. In the parameter ranges considered by Barnard, Richards and Tiedeman, we give a complete characterization of all orders $λ,μ\in\mathbb{R}$ for which $$A_λ(w;1,1-r)\leq H_a(r)\leq A_μ(w;1,1-r)$$ holds for every $r\in(0,1)$. In particular, our results settle completely their two power-mean conjectures. The two sharp orders are determined by the second-order expansion at $r=0$ and the endpoint matching as $r\to1$. A weighted Wronskian identity and a sign analysis of its residual establish the global inequalities.

math.CA↗