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Ilya Zlotnikov

Publications and source records attributed to Ilya Zlotnikov.

16 recordsLinked to original sources
arXiv2026

Asymptotic safety regions for Gabor frames generated by Hermite functions

The aim of this paper is to establish new regions in the frame sets of Hermite functions $h_n$. A classical result of Gröchenig and Lyubarskii shows that the Gabor system $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame for $L^2(\mathbb{R})$ whenever the lattice density exceeds $n+1$. We show that, for every $η>0$ and all sufficiently large $n$, the same Gabor system forms a frame whenever $ab\leq n^{-\frac{2}{3}-η}$. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters $a$ or $b$ is small. Namely, for every $δ>0$ and $ρ\in(0,\frac{1}{2})$ and all sufficiently large $n$ we prove that if $\min\{a,b\}\leq n^{-\frac{1}{2}-δ}$ and $ab\leq \frac{1}{2}-ρ$ then $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame.

math.CA↗
arXiv2026

Periodic Non-uniqueness Sets for Shift-invariant Spaces and Parity-Based Obstructions to the Frame Property for Gabor Systems

The goal of this note is twofold. First, we provide explicit examples of periodic (though not necessarily lattice) sets that give rise to Gabor systems failing to form frames. Our constructions depend only on the parity of the window function $g$. Second, for a wide range of finite-dimensional function spaces $V$ we show that $V$ contains a function $g$ such that a lattice of high density fails to generate a Gabor frame. In particular, we prove that the Gröchenig-Lyubarskii theorem is sharp in the finite-dimensional space of polynomials with Gaussian weight. More precisely, for every $N\in\mathbb{N}$ and every $α,β>0$ satisfying $αβ=\frac{1}{N+1}$, we give an explicit algorithm for finding an even or odd polynomial $p$ of degree at most $N$ such that $\mathcal{G}(p(x)e^{-πx^2}, α\mathbb{Z} \times β\mathbb{Z})$ does not form a frame. The proofs are constructive, elementary, and based on linear algebra.

math.FA↗
arXiv2025

Contractive Hardy--Littlewood inequalities in the Dirichlet range

The class $A_α^p$ consists of those analytic functions $f$ in the unit disc such that \[\|f\|_{α,p}^p := |f(0)|^p+\int_0^1 \left(\frac{d}{dr} M_p^p(r,f)\right) (1-r^2)^{α-1} \,dr < \infty,\] where $M_p^p(r,f)$ is the radial integral mean of $|f|^p$ and $0<α, p <\infty$. For $α>1$, $A_α^p$ is the standard weighted Bergman space, and $A_1^p=H^p$. We consider $A_α^p$ for $0<α<1$ and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between $A_α^p$ and the classical Besov spaces. Our main result is the contractive inequality $\|f\|_{β,q} \leq \|f\|_{α,p}$, valid when $0<α<β<\infty$ and $α/p=β/q$. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author ($1\leq α<β$) and Llinares ($β=1$ and $p=2$). The extension of results from the classical range $1\leq α< \infty$ to the Dirichlet range $0<α<1$ uses arguments relying on analytic continuation.

math.CV↗
arXiv2025

On the frame property of Hermite functions and exploration of their frame sets

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 region". For rectangular lattices and Hermite windows of order 4 and higher, we enlarge this safety region by providing new points on the boundary of this region. For this purpose, we employ the Janssen representation of the frame operator to compare its distance to the identity in the operator norm. The calculations lead to estimates on series involving Laguerre polynomials with Gaussian weight functions.

math.FA↗
arXiv2024

Stability of shifts, interpolation, and crystalline measures

Let $V^p_Γ(\mathcal{G}),1\leq p\leq\infty,$ be the quasi shift-invariant space generated by $Γ$-shifts of a function $\mathcal{G}$, where $Γ\subset\mathbb{R}$ is a separated set. For several large families of generators $\mathcal{G}$, we present necessary and sufficient conditions on $Γ$ that imply that the $Γ$-shifts of $\mathcal{G}$ form an unconditional basis for $V^p_Γ(\mathcal{G})$. The connection between this property, interpolation, universal interpolation, and crystalline measures is discussed.

math.FA↗
arXiv2024

Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials

We introduce two families of generators (functions) $\mathcal{G}$ that consist of entire and meromorphic functions enjoying a certain periodicity property and contain the classical Gaussian and hyperbolic secant generators. Sharp results are proved on the density of separated sets that provide non-uniform sampling for the shift-invariant and quasi shift-invariant spaces generated by elements of these families. As an application, we obtain new sharp results on the density of semi-regular lattices for the Gabor frames generated by elements from these families.

math.FA↗
arXiv2024

Sampling in the shift-invariant space generated by the bivariate Gaussian function

We study the space spanned by the integer shifts of a bivariate Gaussian function and the problem of reconstructing any function in that space from samples scattered across the plane. We identify a large class of lattices, or more generally semi-regular sampling patterns spread along parallel lines, that lead to stable reconstruction while having densities close to the critical value given by Landau's limit. At the critical density, we construct examples of sampling patterns for which reconstruction fails. In the same vein, we also investigate continuous sampling along non-uniformly scattered families of parallel lines and identify the threshold density of line configurations at which reconstruction is possible. In a remarkable contrast with Paley-Wiener spaces, the results are completely different for lines with rational or irrational slopes. Finally, we apply the sampling results to Gabor systems with bivariate Gaussian windows. As a main contribution, we provide a large list of new examples of Gabor frames with non-complex lattices having volume close to 1.

math.FA↗
arXiv2022

Completeness of Certain Exponential Systems and Zeros of Lacunary Polynomials

Let $Γ$ be a subset of $\{0,1,2,...\}$. We show that if $Γ$ has `gaps' then the completeness and frame properties of the system $\{t^ke^{2πi nt}: n\in\mathbb{Z},k\inΓ\}$ differ from those of the classical exponential systems. This phenomenon is closely connected with the existence of certain uniqueness sets for lacunary polynomials.

math.CA↗
arXiv2022

Contractive projections in Paley-Wiener spaces

Let $S_1$ and $S_2$ be disjoint finite unions of parallelepipeds. We describe necessary and sufficient conditions on the sets $S_1,S_2$ and exponents $p$ such that the canonical projection $P$ from $PW_{S_1\cup S_2}^p$ to $PW_{S_1}^p$ is a contraction.

math.FA↗
arXiv2022

On planar sampling with Gaussian kernel in spaces of bandlimited functions

Let $I=(a,b)\times(c,d)\subset {\mathbb R}_{+}^2$ be an index set and let $\{G_α(x) \}_{α\in I}$ be a collection of Gaussian functions, i.e. $G_α(x) = \exp(-α_1 x_1^2 - α_2 x_2^2)$, where $α= (α_1, α_2) \in I, \, x = (x_1, x_2) \in {\mathbb R}^2$. We present a complete description of the uniformly discrete sets $Λ\subset {\mathbb R}^2$ such that every bandlimited signal $f$ admits a stable reconstruction from the samples $\{f \ast G_α (λ)\}_{λ\in Λ}$.

math.CA↗
arXiv2021

Sharp multiplicative inequalities with $\mathrm{BMO}$ $\mathrm{II}$

We find the best possible constant $C$ in the inequality $$\|φ\|_{L^r}^{\phantom{\frac{p}{r}}}\leq C\|φ\|_{L^p}^{\frac{p}{r}}\|φ\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$$ for all possible values of parameters $p$ and $r$ such that $1 \le p < r < +\infty$. We employ the Bellman function technique to solve this problem. The Bellman function of three variables corresponding to this problem has a rather complicated structure, however, we managed to provide the explicit formulas for this function. First, we solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in multi-dimensional cases.

math.CA↗
arXiv2021

On geometry of the unit ball of Paley-Wiener space over two symmetric intervals

Let $PW_S^1$ be the space of integrable functions on $\mathbb{R}$ whose Fourier transform vanishes outside $S$, where $S = [-σ,-ρ]\cup[ρ,σ]$, $0<ρ<σ$. In the case $ρ>σ/2$, we present a complete description of the set of extreme and the set of exposed points of the unit ball of $PW^1_S$. The structure of these sets becomes more complicated when $ρ<σ/2$.

math.FA↗
arXiv2021

Sharp moment estimates for martingales with uniformly bounded square functions

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of the terminate distribution of a martingale whose square function is uniformly bounded by one. We introduce a Bellman function for the corresponding extremal problem and reduce it to the already known Bellman function on $\mathrm{BMO}([0,1])$. In the case of tail estimates, a similar reduction does not work exactly, so we come up with a fine supersolution that leads to sharp tail estimates.

math.PR↗
arXiv2020

On 2-dimensional mobile sampling

Necessary and sufficient conditions are presented for several families of planar curves to form a set of stable sampling for the Bernstein space $\mathcal{B}_Ω$ over a convex set $Ω\subset \mathbb{R}^2$. These conditions "essentially" describe the mobile sampling property of these families for the Paley-Wiener spaces $\mathcal{PW}^p_Ω,1\leq p<\infty$.

math.CA↗