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D. V. Gorbachev

Publications and source records attributed to D. V. Gorbachev.

15 recordsLinked to original sources
arXiv2026

Exact Asymptotics of the Multidimensional Nikolskii Constant

We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant \[ \mathcal L^*(d)=\Bigl(\frac{\pi}{2}+o(1)\Bigr)2^{-d}, \quad d\to\infty. \] In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function $\varphi_d$, and determine their limiting distribution as $d\to\infty$. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent $x^{d+1}\varphi_d(x)$ as the product of two solutions of the equation $u''+V_d(x)u=0$, compare this equation with a Bessel model, and analyze relative canonical products.

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arXiv2026

The Nikolskii Constant in Arbitrary Dimension

We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential type $1/2$ satisfying a functional equation and second-order differential equations with polynomial coefficients. As a result, the original extremal problem is reduced to a one-dimensional spectral problem depending on at most $d+1$ parameters. We also obtain a zeta interpretation of the coefficients of the polynomial appearing in the functional equation and a multiplicative equilibrium condition for the zeros of the extremal function. These results can be used to construct several algorithms for computing $\mathcal C_d$.

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arXiv2026

The Nikolskii constant in odd dimensions

Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\varphi_d(z)=\Phi(z)\Phi(-z)$, and $\Phi$ satisfies a quadratic functional equation and a second-order linear differential equation with polynomial coefficients. Thus, the problem of finding the extremal function is reduced to a spectral problem with finitely many parameters. This result extends a recent one-dimensional result, but uses a different method. For example, in dimension $d=3$ it leads to a seven-diagonal spectral scheme that allows us to compute $\mathcal C_3$ to high accuracy. Even dimensions remain open within this approach.

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arXiv2026

A Bernstein--Ganzburg limit theorem for best weighted approximation

We prove a Bernstein--Ganzburg type limit relation \[ \lim_{n\to\infty} \Bigl(\frac{n}{\sigma}\Bigr)^{(2a+1)/p}E_{n,\sigma}(f)_{p,a,b} =A_{\sigma}(f)_{p,a}, \] where $E_{n,\sigma}(f)_{p,a,b}$ is the error of best approximation of $f(nt/\sigma)$ by trigonometric polynomials of degree at most $n$ in $L^{p}((-\pi,\pi],|2\sin(t/2)|^{2a}|\cos(t/2)|^{2b}\,dt)$, and $A_{\sigma}(f)_{p,a}$ is the error of best approximation of $f$ by entire functions of exponential type at most $\sigma$ in $L^{p}(\mathbb{R},|x|^{2a}\,dx)$. For $a=b=0$, this result was obtained by M.~I.~Ganzburg. The proof uses ideas from the Bernstein--Ganzburg limit theorems and a localization method with the Fej\'er kernel from the proof of the limit relation for Nikol'skii constants. As an application, using known results for polynomial approximation, we compute the exact value of $A_{\pi}(\mathbf{1}_{(-1,1)})_{1,a}$ for $a=0$ and $a=1/2$.

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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.

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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.

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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.

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arXiv2018

Fractional smoothness in $L^p$ with Dunkl weight and its applications

We define fractional power of the Dunkl Laplacian, fractional modulus of smoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight. As application, we prove direct and inverse theorems of approximation theory, and some inequalities for entire functions of spherical exponential type in fractional settings.

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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.

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arXiv2003

Relation between Turán extremum problem and van der Corput sets

Let $K\subset\mathbb N$ and $\mathbf T(K)$ is a set of trigonometric polynomials \[ T(x)=T_0+\sum_{k\in K, k\le H}T_k\cos(2πkx), \qquad H>1, \] $T(x)\ge0$ for all $x$ and $T(0)=1$. Suppose that $0<h\le1/2$ and $K(h)$ is the class of functions \[ f(x)=\sum_{n=0}^{\infty}a_n\cos(2πnx) \] satisfying the following conditions: $a_n\ge0$ for all $n$, $f(0)=1$ and $f(x)=0$ for $h\le|x|\le1/2$. We consider an relation between extremum problem \[ δ(K)=\inf_{T\in\mathbf T(K)}T_0 \] and Turán extremum problem \[ A(h)=\sup_{f\in K(h)}a_0=\sup_{f\in K(h)}\int_{-h}^hf(x) dx \] for rational numbers $h=p/q$ and set $K=\bigcup\limits_{ν=0}^\infty\{qν+p,...,qν+q-p\}$. The problem $δ(K)$ is connection with van der Korput sets. Van der Korput sets study in analytic number theory.

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arXiv2002

Turan Extremum Problem for Periodic Function with Small Support

We consider an extremum problem posed by Turan. The aim of this problem is to find a maximum mean value of 1-periodic continuous even function such that sum of Fourier coefficient modules for this function is equal to 1 and support of this function lies in $[-h,h]$, $0<h\le 1/2$. We show that this extremum problem for rational $h=p/q$ is equivalent two finite-dimensional linear programming problems. Here there are exact results for rational $h=2/q$, $h=p/(2p+1)$, $h=3/q$, and asymptotic equalities.

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