Nikolskii constants for polynomials on the unit sphere

Abstract

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space ${\Pi }_n^d$ of spherical polynomials of degree at most $n$ on the unit sphere ${\mathbb{S}}^d\subset {\mathbb{R}}^{d+1}$ as $n\to \infty$. It is shown that for $0<p<\infty$, \[ \lim_{n\to \infty} \sup\Bigl\{\frac{\|P\|_{L^\infty(\mathbb{S}^d)}}{n^{\frac dp}\|P\|_{L^p(\mathbb{S}^d)}}:\ \ P\inΠ_n^d\Bigr\} =\sup\Bigl\{ \frac{\|f\|_{L^\infty(\mathbb{R}^{d})}}{\|f\|_{L^p(\mathbb{R}^d)}}:\ \ f\in\mathcal{E}_p^d \Bigr\}, \] where ${\mathcal{E}}_p^d$ denotes the space of all entire functions of spherical exponential type at most $1$ whose restrictions to ${\mathbb{R}}^d$ belong to the space $L^p({\mathbb{R}}^d)$, and it is agreed that $0/0=0$. It is further proved that for $0<p<q<\infty$, \[ \liminf_{n\to \infty} \sup\Bigl\{\frac{\|P\|_{L^q(\mathbb{S}^d)}}{n^{d(1/p-1/q)}\|P\|_{L^p(\mathbb{S}^d)}}:\ \ P\inΠ_n^d\Bigr\} \ge \sup\Bigl\{ \frac{\|f\|_{L^q(\mathbb{R}^{d})}}{\|f\|_{L^p(\mathbb{R}^d)}}:\ \ f\in\mathcal{E}_p^d\Bigr\}. \] These results extend the recent results of Levin and Lubinsky for trigonometric polynomials on the unit circle. The paper also determines the exact value of the Nikolskii constant for nonnegative functions with $p=1$ and $q=\infty$: $$\underset{n\to \infty }{\lim }\underset{0\le P\in {\Pi }_n^d}{\sup }\frac{\Vert P{\Vert }_{L^{\infty }({\mathbb{S}}^d)}}{\Vert P{\Vert }_{L^1({\mathbb{S}}^d)}}=\underset{0\le f\in {\mathcal{E}}_1^d}{\sup }\frac{\Vert f{\Vert }_{L^{\infty }({\mathbb{R}}^d)}}{\Vert f{\Vert }_{L^1({\mathbb{R}}^d)}}=\frac{1}{4^d{\pi }^{d/2}\Gamma (d/2+1)}.$$

21 pages