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S. Yu. Tikhonov

Publications and source records attributed to S. Yu. Tikhonov.

9 recordsLinked to original sources
arXiv2025

Chebyshev systems and Sturm oscillation theory for discrete polynomials

We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $\Phi_n=\{\varphi_k\}_{k=1}^n$ on the interval $[0,q]_{\scriptscriptstyle\mathbb{Z}}=[0,q]\cap \mathbb{Z}$. In particular, we establish that the polynomial of best uniform approximation of a discrete function admits a Chebyshev alternance set of length $n+1$ if and only if $\Phi_n$ is a Chebyshev $T_{\scriptscriptstyle\mathbb{Z}}$-system. We also obtain a discrete version of Sturm's oscillation theorem, according to which the number of discrete zeros of the polynomial $\sum_{k=m}^{n}a_k\varphi_k$ is no less than $m-1$ and no more than $n-1$. This implies that $\Phi_n$ is a $T_{\scriptscriptstyle\mathbb{Z}}$-system and a discrete Sturm-Hurwitz spectral gap theorem is valid. As applications, we study the orthogonal polynomials with removed largest zeros. We~establish the monotonicity property of coefficients in the Fourier expansions of such polynomials, thereby strengthening the results of H.~Cohn and A.~Kumar. We apply this to solve a Yudin-type extremal problem for polynomials with spectral gap.

math.CA↗
arXiv2022

On the kernel of the $(κ,a)$-generalized Fourier transform

For the kernel $B_{κ,a}(x,y)$ of the $(κ,a)$-generalized Fourier transform $\mathcal{F}_{κ,a}$, acting in $L^{2}(\mathbb{R}^{d})$ with the weight $|x|^{a-2}v_κ(x)$, where $v_κ$ is the Dunkl weight, we study the important question of when $\|B_{κ,a}\|_{\infty}=B_{κ,a}(0,0)=1$. The positive answer was known for $d\ge 2$ and $\frac{2}{a}\in\mathbb{N}$. We investigate the case $d=1$ and $\frac{2}{a}\in\mathbb{N}$. Moreover, we give sufficient conditions on parameters for $\|B_{κ,a}\|_{\infty}>1$ to hold with $d\ge 1$ and any $a$. We also study the image of the Schwartz space under the $\mathcal{F}_{κ,a}$ transform. In particular, we obtain that $\mathcal{F}_{κ,a}(\mathcal{S}(\mathbb{R}^d))=\mathcal{S}(\mathbb{R}^d)$ only if $a=2$. Finally, extending the Dunkl transform, we introduce non-deformed transforms generated by $\mathcal{F}_{κ,a}$ and study their main properties.

math.CA↗
arXiv2022

Logan's problem for Jacobi transform

We consider direct and inverse Jacobi transforms with measures $dμ(t)=2^{2ρ}(\sinh t)^{2α+1}(\cosh t)^{2β+1}\,dt$ and $dσ(λ)=(2π)^{-1}\bigl|\frac{2^{ρ-iλ}Γ(α+1)Γ(iλ)} {Γ((ρ+iλ)/2)Γ((ρ+iλ)/2-β)}\bigr|^{-2}\,dλ$, respectively. We solve the following generalized Logan problem: to find \[ \infΛ((-1)^{m-1}f), \quad m\in \mathbb{N}, \] where $Λ(f)=\sup\,\{λ>0\colon f(λ)>0\}$ and the infimum is taken over all nontrivial even entire functions $f$ of exponential type that are Jacobi transforms of positive measures with supports on an interval. Here, if $m\ge 2$, then we additionally assume that $\int_{0}^{\infty}λ^{2k}f(λ)\,dσ(λ)=0$ for $k=0,\dots,m-2$. We prove that admissible functions for this problem are positive definite with respect to the inverse Jacobi transform. The solution of Logan's problem was known only when $α=β=-1/2$. We find a unique (up to multiplication by a positive constant) extremizer $f_m$. The corresponding Logan problem for the Fourier transform on the hyperboloid $\mathbb{H}^{d}$ is also solved. Using properties of the extremizer $f_m$ allows us to give an upper estimate of the length of a minimal interval containing not less than $n$ zeros of positive definite functions. Finally, we show that the Jacobi functions form the Chebyshev systems.

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arXiv2020

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

In this paper we study direct and inverse approximation inequalities in $L^{p}(\mathbb{R}^{d})$, $1<p<\infty$, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function $f$ via the fractional powers of the Dunkl Laplacian of approximants of $f$. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier--Dunkl inequalities are derived.

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arXiv2019

Uncertainty principles for eventually constant sign bandlimited functions

We study the uncertainty principles related to the generalized Logan problem in $\mathbb{R}^{d}$. Our main result provides the complete solution of the following problem: for a fixed $m\in \mathbb{Z}_{+}$, find \[ \sup\{|x|\colon (-1)^{m}f(x)>0\}\cdot \sup \{|x|\colon x\in \mathrm{supp}\,\widehat{f}\,\}\to \inf, \] where the infimum is taken over all nontrivial positive definite bandlimited functions such that $\int_{\mathbb{R}^d}|x|^{2k}f(x)\,dx=0$ for $k=0,\dots,m-1$ if $m\ge 1$. We also obtain the uncertainty principle for bandlimited functions related to the recent result by Bourgain, Clozel, and Kahane.

math.CA↗
arXiv2018

Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications

We prove that the spherical mean value of the Dunkl-type generalized translation operator $τ^y$ is a positive $L^p$-bounded generalized translation operator $T^t$. As application, we prove the Young inequality for a convolution defined by $T^t$, the $L^p$-boundedness of $τ^y$ on a radial functions for $p>2$, the $L^p$-boundedness of the Riesz potential for the Dunkl transform and direct and inverse theorems of approximation theory in $L^p$-spaces with the Dunkl weight.

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arXiv2017

Riesz potential and maximal function for Dunkl transform

We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev type inequality and weighted week $(L^1, L^q)$ estimate. We find a sharp constant in the weighted $L^p$-inequality, generalizing the results of W. Beckner and S. Samko.

math.CA↗
arXiv2013

Mixed Moduli of Smoothness in $L_p$, $1<p<\infty$

In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from $L_p$, $1<p<\infty$. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed.

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