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This article is cited in 1 scientific paper (total in 1 paper)
Nikol'skii constants for compact homogeneous spaces
D. V. Gorbachevab a N. N. Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg)
b Tula State University (Tula)
Abstract:
In this paper, we study the sharp Lp-Nikol'skii constants for the case of Riemannian symmetric manifolds Md of rank 1. These spaces are fully classified and include the unit Euclidean sphere Sd, as well as the projective spaces Pd(R), Pd(C), Pd(H), P16(Ca). There is a common harmonic analysis on these manifolds, in particular, the subspaces of polynomials Πn(Md) of order at most n are defined. In the general case, the sharp Lp-Nikol'skii constant for the subspace Y⊂L∞ is defined by the equality
C(Y,Lp)=supf∈(Y∩Lp)∖{0}‖f‖∞‖f‖p.
V.A. Ivanov (1983) gave the asymptotics
C(Πn(Md),Lp(Md))≍nd/p,n→∞,p∈[1,∞).
For the case of a sphere, this result was significantly improved by the author together with F. Dai and S. Tikhonov (2020):
C(Πn(Sd),Lp(Sd))=C(Ed1,Lp(Rd))nd/p(1+o(1)),n→∞,p∈(0,∞),
where Ed1 is the set of entire functions of exponential spherical type at most 1 bounded on Rd. M.I. Ganzburg (2020) transferred this equality to the case of the multidimensional torus Td and trigonometric polynomials. For d=1, these results follow from the fundamental work of E. Levin and D. Lubinsky (2015). In a joint work of the author and I.A. Martyanov (2020), the following explicit boundaries of the spherical Nikol'skii constant were proved, which refine the above results for p⩾:
n^{d/p}\le \frac{\mathcal{C}(\Pi_{n}(\mathbb{S}^{d}),L^{p}(\mathbb{S}^{d}))} {\mathcal{C}(\mathcal{E}_{1}^{d},L^{p}(\mathbb{R}^{d}))}\le \bigl(n+2\lceil \tfrac{d+1}{2p}\rceil\bigr)^{d/p}, n\in \mathbb{Z}_{+}, p\in [1,\infty).
This result was proved using a one-dimensional version of the problem for the case of a periodic Gegenbauer weight. The development of this method allows us to prove the following general result: for p\ge 1
n^{d/p}\le \frac{\mathcal{C}(\Pi_{n}(\mathbb{M}^{d}),L^{p}(\mathbb{M}^{d}))} {\mathcal{C}(\mathcal{E}_{1}^{d},L^{p}(\mathbb{R}^{d}))}\le \bigl(n+\lceil \tfrac{\alpha_{d}+3/2}{p}\rceil+\lceil \tfrac{\beta_{d}+1/2}{p}\rceil\bigr)^{d/p},
where \alpha_{d}=d/2-1, \beta_{d}=d/2-1, -1/2, 0, 1, 3 respectively for \mathbb{S}^{d}, \mathbb{P}^{d}(\mathbb{R}), \mathbb{P}^{d}(\mathbb{C}), \mathbb{P}^{d}(\mathbb{H}), \mathbb{P}^{16}(\mathrm {Ca}). The proof of this result is based on the connection of harmonic analysis on \mathbb{M}^{d} with Jacobi analysis on [0,\pi] and \mathbb{T} with periodic weight \bigl|2\sin \tfrac{t}{2}\bigr|^{2\alpha+1}\bigl|\cos \tfrac{t}{2}\bigr|^{2\beta+1}. Also we give related results for the trigonometric Nikol'skii constants in L^{p} on \mathbb{T} with Jacobi weight and Nikol'skii constants for entire functions of exponential type in L^{p} on \mathbb{R} with power weight.
Keywords:
Nikolskii constant, homogeneous space, polynomial, entire function of exponential type, Jacobi weight.
Received: 27.08.2021 Accepted: 06.12.2021
Citation:
D. V. Gorbachev, “Nikol'skii constants for compact homogeneous spaces”, Chebyshevskii Sb., 22:4 (2021), 100–113
Linking options:
https://www.mathnet.ru/eng/cheb1095 https://www.mathnet.ru/eng/cheb/v22/i4/p100
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