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Nikol'skii–Bernstein Constants for Entire Functions of Exponential Spherical Type in Weighted Spaces
D. V. Gorbachev,
V. I. Ivanov Tula State University
Abstract:
We study the exact constant in the Nikol'skii–Bernstein inequality
$\|Df\|_{q}\le C\|f\|_{p}$ on the subspace of entire functions
$f$ of exponential spherical type in the space
$L^{p}(\mathbb{R}^{d})$ with a power-type weight
$v_{\kappa}$. For the differential operator
$D$, we take a nonnegative integer power of the Dunkl Laplacian
$\Delta_{\kappa}$ associated with the weight
$v_{\kappa}$. This situation encompasses the one-dimensional case of the space
$L^{p}(\mathbb{R}_{+})$ with the power weight
$t^{2\alpha+1}$ and Bessel differential operator. Our main result consists in the proof of an equality between the multidimensional and one-dimensional weighted constants for
$1\le p\le q=\infty$. For this, we show that the norm
$\|Df\|_{\infty}$ can be replaced by the value
$Df(0)$, which was known only in the one-dimensional case. The required mapping of the subspace of functions, which actually reduces the problem to the radial and, hence, one-dimensional case, is implemented by means of the positive operator of Dunkl generalized translation
$T_{\kappa}^{t}$. We prove its new property of analytic continuation in the variable
$t$. As a consequence, we calculate the weighted Bernstein constant for
$p=q=\infty$, which was known in exceptional cases only. We also find some estimates of the constants and give a short list of open problems.
Keywords:
Nikol'skii–Bernstein inequality, exact constant, entire function of exponential spherical type, power-type weight, Dunkl Laplacian.
UDC:
517.5
MSC: 41A17,
42B10 Received: 08.04.2019
DOI:
10.21538/0134-4889-2019-25-2-75-87
Fulltext:
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