Nikol'skii constants for compact homogeneous spaces

D. V. Gorbachev

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 100-113.

Résumé

In this paper, we study the sharp $L^{p}$-Nikol'skii constants for the case of Riemannian symmetric manifolds $\mathbb{M}^{d}$ of rank $1$. These spaces are fully classified and include the unit Euclidean sphere $\mathbb{S}^{d}$, as well as the projective spaces $\mathbb{P}^{d}(\mathbb{R})$, $\mathbb{P}^{d}(\mathbb{C})$, $\mathbb{P}^{d}(\mathbb{H})$, $\mathbb{P}^{16}(\mathrm{Ca})$. There is a common harmonic analysis on these manifolds, in particular, the subspaces of polynomials $\Pi_{n}(\mathbb{M}^{d})$ of order at most $n$ are defined. In the general case, the sharp $ L^{p}$-Nikol'skii constant for the subspace $Y\subset L^{\infty}$ is defined by the equality $$ \mathcal{C}(Y,L^{p})=\sup_{f\in (Y\cap L^{p})\setminus \{0\}}\frac{\|f\|_{\infty}}{\|f\|_{p}}. $$ V.A. Ivanov (1983) gave the asymptotics $$ \mathcal{C}(\Pi_{n}(\mathbb{M}^{d}),L^{p}(\mathbb{M}^{d}))\asymp n^{d/p}, n\to \infty, p\in [1,\infty). $$ For the case of a sphere, this result was significantly improved by the author together with F. Dai and S. Tikhonov (2020): $$ \mathcal{C}(\Pi_{n}(\mathbb{S}^{d}),L^{p}(\mathbb{S}^{d}))= \mathcal{C}(\mathcal{E}_{1}^{d},L^{p}(\mathbb{R}^{d}))n^{d/p}(1+o(1)), n\to \infty, p\in (0,\infty), $$ where $\mathcal{E}_{1}^{d}$ is the set of entire functions of exponential spherical type at most $1$ bounded on $\mathbb{R}^{d}$. M.I. Ganzburg (2020) transferred this equality to the case of the multidimensional torus $\mathbb{T}^{d}$ 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 $p\ge 1$: $$ n^{d/p}\le \frac{\mathcal{C}(\Pi_{n}(\mathbb{S}^{d}),L^{p}(\mathbb{S}^{d}))} {\mathcal{C}(\mathcal{E}_{1}^{d},L^{p}(\mathbb{R}^{d}))}\le \bigl(n+2\lceil \tfrac{d+1}{2p}\rceil\bigr)^{d/p}, n\in \mathbb{Z}_{+}, p\in [1,\infty). $$ 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.

Détail

Keywords: Nikolskii constant, homogeneous space, polynomial, entire function of exponential type, Jacobi weight.

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