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Chebyshevskii Sbornik, 2021, Volume 22, Issue 5, Pages 161–171
DOI: https://doi.org/10.22405/2226-8383-2021-22-5-161-171
(Mi cheb1124)
 

This article is cited in 1 scientific paper (total in 1 paper)

On an extremal problem for positive definite functions

A. D. Manov

Donetsk National University (Donetsk)
Full-text PDF (605 kB) Citations (1)
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Abstract: In this paper we consider an extremal problem related to a set of continuous positive definite functions on $\mathbb{R}$ whose support is contained in the closed interval $[-\sigma,\sigma]$, $\sigma>0$ and the value at the origin is fixed (the class $\mathfrak{F}_\sigma$).
We consider the following problem. Let $\mu$ be a linear locally bounded functional on the set of continuous functions which have compact support, i.e. $C_c(\mathbb{R})$ and suppose that $\mu$ is real-valued functional on the sets $\mathfrak{F}_\sigma$, $\sigma>0$. For a fixed $\sigma>0$, it is required to find the following constants:
$$ M(\mu,\sigma):=\sup\left\{ \mu(\varphi): \varphi\in\mathfrak{F}_\sigma\right\},\ m(\mu,\sigma):=\inf\left\{ \mu(\varphi): \varphi\in\mathfrak{F}_\sigma\right\}. $$
We have obtained a general solution to this problem for functionals of the following form $\mu(\varphi)=\int_\mathbb{R}\varphi(x)\rho(x)dx$, $\varphi\in C_c(\mathbb{R})$, where $\rho\in L_{loc}(\mathbb{R})$ and $\rho(x)=\overline{\rho(-x)}$ a.e. on $x\in\mathbb{R}$. For $\rho(x)\equiv1$, the value of $M(\mu,\sigma)$ was obtained by Siegel in 1935 and, independently, by Boas and Kac in 1945. In this article, we have obtained explicit solution to the problem under consideration in cases of $\rho(x)=ix$, $\rho(x)=x^2$ and $\rho(x)=i\mathop{\rm sign} x$, $x\in\mathbb{R}$.
In addition, in this paper we study the connection between the problem under consideration and pointwise inequalities for entire functions of exponential type $\leqslant\sigma$ whose restrictions on $\mathbb{R}$ are in $L_1(\mathbb{R})$. In particular, sharp inequalities are obtained for the first and second derivatives of such functions.
Keywords: positive-definite functions, extremal problems, Bochner theorem, Fourier transform, entire functions of exponential type.
Received: 17.06.2021
Accepted: 21.12.2021
Document Type: Article
UDC: 517.5+519.213
Language: Russian
Citation: A. D. Manov, “On an extremal problem for positive definite functions”, Chebyshevskii Sb., 22:5 (2021), 161–171
Citation in format AMSBIB
\Bibitem{Man21}
\by A.~D.~Manov
\paper On an extremal problem for positive definite functions
\jour Chebyshevskii Sb.
\yr 2021
\vol 22
\issue 5
\pages 161--171
\mathnet{http://mi.mathnet.ru/cheb1124}
\crossref{https://doi.org/10.22405/2226-8383-2021-22-5-161-171}
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  • https://www.mathnet.ru/eng/cheb/v22/i5/p161
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    References:29
     
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