arXiv subjects

Sergey Tikhonov

Publications and source records attributed to Sergey Tikhonov.

At least 19 recordsLinked to original sources

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 $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$.

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arXiv2026

A Logvinenko-Sereda theorem for lacunary spectra

For a function $F$ represented as $F(x)=\sum_{n=0}^\infty{f_n (x) e^{2 πi λ_n x}},$ where each $f_n$ satisfies $\operatorname{spec}(f_n) \subset [0, 1]$ and $(λ_n)_{n\geq 0}\subset \mathbb{R}_+$ is a lacunary sequence, we obtain $$ \|F\|_{L^2(\mathbb{R})}\lesssim \|Fχ_{E}\|_{L^2(\mathbb{R})} $$ provided that $E$ is a thick subset of $\mathbb{R}$. This extends the Logvinenko-Sereda theorem and answers a question posed by Kovrizhkin for functions with positive frequencies.

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arXiv2026

Heisenberg's inequality in $L^p$

In this paper, we obtain non-symmetric and symmetric versions of the classical Heisenberg-Pauli-Weyl uncertainty principle in Lebesgue spaces with power weights.

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arXiv2025

Extremizers of a Fourier uncertainty principle related to averaging

We study the uncertainty principle $$\lVert\widehatμ(ξ) |ξ|^β\rVert_\infty^α \left(\int |x|^αd μ\right)^β \geq C(α,β,d){\lVertμ\rVert_{TV}^{α+β}}$$ for finite non-negative measures on $\mathbb{R}^d $. We prove that $C(α,β,d)>0$ for all $α,β>0$ and that extremizers exist. Moreover, we obtain an abstract characterization of the extremizers, which allows us to describe their asymptotic behavior and, for certain parameter values, to determine them explicitly.

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arXiv2025

Bernstein-Nikolskii Inequality: Optimality with Respect to the Smoothness Parameter

In this paper, we study the form of the constant $C$ in the Bernstein--Nikolskii inequalities $\|f^{(s)}\|_q \lesssim C(s, p, q)\left\|f\right\|_p,\,0<p<q \leq\infty$, for trigonometric polynomials and entire functions of exponential type. We obtain the optimal behavior of the constant with respect to the smoothness parameter $s$.

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arXiv2025

Kahane-Katznelson-de Leeuw theorem and absolute convergence of Fourier series

We extend the Kahane-Katznelson-de Leeuw theorem to smoothness spaces by showing that for any $g \in W^{l,2}(\mathbb{T}^d)$, there exists a function $f\in C^l(\mathbb{T}^d)$ satisfying $|\widehat{f}(n)|\geq |\widehat{g}(n)|$ and $$ω_r(D^l f,t)_\infty \approx ω_r(D^l g,t)_2, \quad t>0. $$ We apply this result to solve the Bernstein problem of finding necessary and sufficient conditions for the absolute convergence of multiple Fourier series. Finally, we explore the absolute integrability of Fourier transforms.

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arXiv2025

The Fourier transform is an extremizer of a class of bounded operators

We show that, for a natural class of rearrangement admissible spaces $X$ and $Y$, the Fourier operator is bounded between $X$ and $Y$ if and only if any operator of joint strong type $(1,\infty; 2,2)$ is also bounded between $X$ and $Y$. By using this result, we fully characterize the weighted Fourier inequalities of the form $$\qquad\qquad \lVert\widehat{f}u \rVert_q \leq C \lVert fv\rVert_p,\quad 1\leq p\leq \infty,\,0<q\leq \infty,$$ for radially monotone weights $(u,v)$. This answers a long-standing problem posed by Benedetto-Heinig, Jurkat-Sampson, and Muckenhoupt. In the case of $p\le q$, such a characterization has been known since the 1980s.

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arXiv2024

Poisson summation formula in weighted Lebesgue spaces

We characterize the parameters $(α,β,p,q)$ for which the condition $f|x|^α\in L^p$ and $\widehat{f}|ξ|^β\in L^q$ implies the validity of the Poisson summation formula, thus completing the study of Kahane and Lemarié-Rieusset.

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arXiv2024

Marcinkiewicz-Zygmund inequalities in quasi-Banach function spaces

We obtain Marcinkiewicz--ygmund (MZ) inequalities in various Banach and quasi-Banach spaces under minimal assumptions on the structural properties of these spaces. Our main results show that the Bernstein inequality in a general quasi-Banach function lattice $X$ implies Marcinkiewicz-Zygmund type estimates in $X$. We present a general approach to obtain MZ inequalities not only for polynomials but for other function classes including entire functions of exponential type, splines, exponential sums, etc.

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arXiv2024

Sampling discretization in Orlicz spaces

We obtain new sampling discretization results in Orlicz norms on finite dimensional spaces. As applications, we study sampling recovery problems, where the error of the recovery process is calculated with respect to different Orlicz norms. In particular, we are interested in the recovery by linear methods in the norms close to $L^2$.

math.FA↗

arXiv2024

Subcritical Fourier uncertainty principles

It is well known that if a function $f$ satisfies $$\|f(x) e^{πα|x|^2}\|_p + \| \widehat{f}(ξ) e^{πα|ξ|^2} \|_q<\infty \qquad\qquad\qquad(*)$$ with $α=1$ and $1\le p,q<\infty$, then $f\equiv 0.$ We prove that if $f$ satisfies $(*)$ with some $0<α<1$ and $1\le p,q\leq \infty$, then $$ |f(y)|\le C (1+|y|)^{\frac{d}{p}} e^{- πα|y|^2}, \quad y\in \mathbb{R}^d, $$ with $ C=C(α,d,p,q)$ and this bound is sharp for $p\neq 1$. We also study a related uncertainty principle for functions satisfying $\;\;\displaystyle\|f(x)|x|^m\|_p+ \|\widehat{f}(ξ)|ξ|^n\|_q <\infty.$

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arXiv2024

Note on Fourier inequalities

We prove that the Hausdorff--Young inequality $\|{\widehat{f}}\|_{q(\cdot)} \leq C \|{f}\|_{p(\cdot)}$ with $q(x)=p'(1/x)$ and $p(\cdot)$ even and non-decreasing holds in variable Lebesgue spaces if and only if $p$ is a constant. However, under the additional condition on monotonicity of $f$, we obtain a full characterization of Pitt-type weighted Fourier inequalities in the classical and variable Lebesgue setting.

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arXiv2024

New approach to affine Moser-Trudinger inequalities via Besov polar projection bodies

We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in $W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to the remaining range $p \geq n/s$. For each value of $s$, our results are stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct, we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the affine invariant provided by Besov polar projection bodies.

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arXiv2024

A unified approach to inequalities for K-functionals and moduli of smoothness

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

math.FA↗

arXiv2023

Fourier inequalities in Morrey and Campanato spaces

We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} \|\widehat f\|_{X_{p,q}^λ} \lesssim \|f\|_{Y}, \end{equation} where $X$ is either a Morrey or Campanato space and $Y$ is an appropriate function space. In the case of the Morrey space we sharpen the estimate $ \|\widehat f\|_{M_{p,q}^λ} \lesssim \|f\|_{L_{s',q}},$ $ s\geq 2,$ $\frac{1}{s} = \frac{1}{p}-\fracλ{n}.$ We also show that \eqref{introduction} does not hold when both $X$ and $Y$ are Morrey spaces. If $X$ is a Campanato space, we prove that \eqref{introduction} holds for $Y$ being the truncated Lebesgue space.

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arXiv2023

A unified approach to self-improving property via K-functionals

In this paper we obtain new quantitative estimates that improve the classical inequalities: Poincaré-Ponce, Gaussian Sobolev, and John-Nirenberg. Our method is based on the K-functionals and allows one to derive self-improving type inequalities. We show the optimality of the method by obtaining new Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limiting formulas. In particular, we derive these formulas for fractional powers of infinitesimal generators of operator semigroups on Banach spaces.

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arXiv2022

Truncated smooth function spaces

We introduce truncated Besov and Triebel--Lizorkin function spaces and investigate their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allow us to improve several known results in functional analysis and PDE's.

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arXiv2022

Hardy-Littlewood-type theorems for Fourier transforms in $\R^d$

We obtain Fourier inequalities in the weighted $L_p$ spaces for any $1<p<\infty$ involving the Hardy-Cesàro and Hardy-Bellman operators. We extend these results to product Hardy spaces for $p\le 1$. Moreover, boundedness of the Hardy-Cesàro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. One of our main tools is an appropriate version of the Hardy-Littlewood-Paley inequality $ \|\widehat{f}\|_{L_{p',q}} \lesssim \left\|f\right\|_{L_{p,q}}$.

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