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Aleksei Kulikov

Publications and source records attributed to Aleksei Kulikov.

At least 19 recordsLinked to original sources
arXiv2026

When a meromorphic function that omits three values is of bounded type

Suppose that a function $F$ is meromorphic in the domain $\mathbb H(-m) = \{ z : \mathrm{Im}\, z > -m(\mathrm{Re}\, z) \}$, where $m$ is an even, positive, and continuous function that does not increase on $\mathbb R_{\ge 0}$, and suppose that $F$ omits there three distinct values. Then $F$ is of bounded type in the upper half-plane (i.e., is represented there as a quotient of two bounded analytic functions), provided that the logarithmic integral of the function $m$ is convergent. On the other hand, if the logarithmic integral of $m$ diverges, there exists a function $F$ meromorphic in $\mathbb H(-m)$, that omits there three distinct values, and which is of unbounded type in the upper half-plane. This result is motivated by a century old question originating with Rolf Nevanlinna.

math.CV↗
arXiv2026

Fourier coefficients of continuous functions with sparse spectrum

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in C(\mathbb{T})$ such that $\hat{f}(2^k) = a_k$ and $\hat{f}(n) = 0$ for $n\notin \cup_k [2^k-r_k,2^k+r_k]$? We show that this is possible if and only if $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$.

math.CA↗
arXiv2026

Sharp estimates for eigenvalues of localization operators with applications to area laws

We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large parameter $c > 0$. For the counting function of the eigenvalues $|\{n: \varepsilon < λ_n(A,B)\le 1-\varepsilon\}|$ we obtain a sharp uniform upper bound if one of the sets is a finite disjoint union of parallelepipeds and a bound which is only a single logarithm off the conjectural optimal bound in the general case. These bounds are applied to the estimation of traces ${\rm{Tr}}\, f(S_{A,B})$ for functions $f$ with a very low regularity, in particular establishing an enhanced area law in the former case.

math.SP↗
arXiv2026

Sharp estimates for eigenvalues of localization operators before the plunge region

We study two closely related yet different localization operators: the time-frequency localization operator to the pair of intervals $S_{I, J} = P_I \mathcal{F}^{-1} P_J\mathcal{F} P_I$ and the localization of the coherent state transform to the square $L_Q$. Eigenvalues of both of them exhibit the same phase transition: if $|I| |J| = |Q| = c$ then first $\approx c$ eigenvalues are very close to $1$, then there are $o(c)$ intermediate eigenvalues and the rest of the eigenvalues are very close to $0$. Moreover, for both of them if $n < (1-\varepsilon)c$ for fixed $\varepsilon > 0$ then the eigenvalues are exponentially close to $1$. The goal of this paper is to establish sharp uniform bounds on these eigenvalues when $n$ is close to $c$ and see if there is a qualitative difference between the spectrums of $S_{I, J}$ and $S_Q$. We show that for $n < c -c^{0.99}$, say, in the time-frequency localization case we have $-\log(1-λ_n(c))\asymp\frac{c-n}{\log(\frac{2c}{c-n})}$ while in the coherent state transform case we have $-\log(1-μ_n(c))\asymp (\sqrt{c}-\sqrt{n})^2,$ which is much smaller if $c-n = o(c)$, so there is indeed a difference between these two cases. The proofs crucially rely on the complex-analytic interpretations of these localization operators.

math.CA↗
arXiv2025

Frames for compactly supported functions with irrational density

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;α,β)=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $α< b-a, αβ< 1, αβ\notin\Q$. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions $g$ which give us a frame e.g. $g(x) = \exp(\frac{1}{x^4-1})χ_{(-1,1)}(x)$.

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

Gabor frames for functions supported on a semi-axis

Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on $\mathbb{R}_+$ such that for all $t > 0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system $$\mathcal{G}(g;α,β):=\{g_{m,n}\}_{m,n\in\mathbb{Z}}=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}$$ always forms a frame in $L^2(\mathbb{R})$ for all lattice parameters $α$,$β$, $αβ\leq 1$.

math.FA↗
arXiv2024

Fekete's lemma in Banach spaces

For a sequence of vectors $\{v_n\}_{n\in\mathbb{N}}$ in the uniformly convex Banach space $X$ which for all $n, m\in \mathbb{N}$ satisfy $\|v_{n+m}\|\le \|v_n + v_m\|$ we show the existence of the limit $\lim_{n\to \infty} \frac{v_n}{n}$. This extends the classical Fekete's subadditivite lemma to Banach space-valued sequences.

math.FA↗
arXiv2024

Openness of the frame set on the hyperbolas

We prove that for the functions of the form $g(x) = h(x) + \frac{C}{x+i}$, where $h$ belongs to the continuous Wiener algebra $W_0$, the intersection of the frame set $\mathcal{F}_g$ with every hyperbola $\{α, β> 0 \mid αβ= c\}$ is open in the relative topology. In particular, this applies to all rational functions $g$.

math.FA↗
arXiv2023

Fourier uniqueness and non-uniqueness pairs

Motivated by recent works by Radchenko and Viazovska and by Ramos and Sousa, we find sufficient conditions for a pair of discrete subsets of the real line to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are close to each other. The uniqueness result can be upgraded to an interpolation formula, which in turn produces an abundance of discrete measures with discrete Fourier transform.

math.CA↗
arXiv2023

Exponential lower bound for the eigenvalues of the time-frequency localization operator before the plunge region

We prove that the eigenvalues $λ_n(c)$ of the time-frequency localization operator satisfy $λ_n(c) > 1 - δ^c$ for $n = [(1-\varepsilon)c]$, where $δ= δ(\varepsilon) < 1$ and $\varepsilon > 0$ is arbitrary, improving on the result of Bonami, Jaming and Karoui, who proved it for $\varepsilon \ge 0.42$. The proof is based on the properties of the Bargmann transform.

math.CA↗
arXiv2023

A monotonicity theorem for subharmonic functions on manifolds

We provide a sharp monotonicity theorem about the distribution of subharmonic functions on manifolds, which can be regarded as a new, measure theoretic form of the uncertainty principle. As an illustration of the scope of this result, we deduce contractivity estimates for analytic functions on the Riemann sphere, the complex plane and the Poincaré disc, with a complete description of the extremal functions, hence providing a unified and illuminating perspective of a number of results and conjectures on this subject, in particular on the Wehrl entropy conjecture by Lieb and Solovej. In this connection, we completely prove that conjecture for SU(2), by showing that the corresponding extremals are only the coherent states. Also, we show that the above (global) estimates admit a local counterpart and in all cases we characterize also the extremal subsets, among those of fixed assigned measure.

math.CA↗
arXiv2022

Gabor frame operator for the Cauchy kernel

We obtain frame bounds estimates and the Gabor frame operator $S=S^{α,β}$ for Gabor frames generated by the Cauchy kernel. In addition we find the explicit expression for the canonical dual window for all values of the lattice parameters $α,β$, $αβ\leq 1$.

math.CV↗
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

On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles

We make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schrödinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question.

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

Functionals with extrema at reproducing kernels

We show that certain monotone functionals on the Hardy spaces and convex functionals on the Bergman spaces are maximized at the normalized reproducing kernels among the functions of norm $1$, thus proving the contractivity conjecture of Pavlović and of Brevig, Ortega-Cerdà, Seip and Zhao and the Wehrl-type entropy conjecture for the $SU(1,1)$ group of Lieb and Solovej, respectively.

math.CV↗