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This article is cited in 1 scientific paper (total in 1 paper)
BRIEF MESSAGE
Extremal Nikolskii–Bernstein- and Turán-type problems for Dunkl transform
D. V. Gorbacheva, N. N. Dobrovolskiiab a Tula State University (Tula)
b Tula State Pedagogical University (Tula)
Abstract:
We study the interrelation between the extremal Turán-type problems and
Nikolskii – Bernstein problems for nonnegative functions on $\mathbb{R}^{d}$
with the Dunkl weight. The Turán problem is to find the supremum of a given
moment of a positive definite (with respect to the Dunkl transform) function
with a support in the Euclidean ball and a fixed value at zero. In the sharp
$L^{1}$-Nikolskii–Bernstein inequality, the supremum norm of the Dankl
Laplacian of an entire function of exponential spherical type with the unit
$L^{1}$-norm is estimated. Extremal Feuér and Beaumann problems is also
mentioned. The Dunkl transform covers the case of the classical Fourier
transform in the case of unit weight.
Nikolskii–Bernstein inequalities are classical in approximation theory, and
the Turán-type problems have applications in metric geometry. Nevertheless,
we prove that they have the same answer, which is given explicitly. The easy
proof is relied on our old results from the theory of solving extremal problems
to the Dunkl transform.
Keywords:
Dunkl weight, Fourier–Dunkl transform, entire function of exponential spherical type, positive definite function, Nikolskii–Bernstein constant, Turán extremal problem.
Received: 05.09.2019 Accepted: 12.11.2019
Citation:
D. V. Gorbachev, N. N. Dobrovolskii, “Extremal Nikolskii–Bernstein- and Turán-type problems for Dunkl transform”, Chebyshevskii Sb., 20:3 (2019), 394–400
Linking options:
https://www.mathnet.ru/eng/cheb820 https://www.mathnet.ru/eng/cheb/v20/i3/p394
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Abstract page: | 192 | Full-text PDF : | 60 | References: | 13 |
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