Abstract:
We study the asymptotic behavior of sharp Nikolskii constant
$$\mathcal{C}(n, d, p, q) := \sup \{\|{f}\|_{L^q(\mathbb{S}^d)}: f \in \Pi_n^d, \|{f}\|_{L^p(\mathbb{S}^d)} = 1\}$$
for $0 < p < q \le \infty$ as $n \to \infty$, where $\Pi_n^d$ denotes the space of all spherical polynomials $f$ of degree at most
$n$ on the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$.
1. We prove that for $0 < p < \infty$ and $q = \infty$,
$$\lim\limits_{n \to \infty}
\frac{\mathcal{C}(n, d, p, \infty)}
{n^{d/p}} = \mathcal{L}(d, p, \infty),$$
and for $0 < p < q < \infty$,
$$\liminf\limits_{n \to \infty}
\frac{\mathcal{C}(n, d, p, q)}
{n^{d (1/p-1/q)}}
\ge \mathcal{L}(d, p, q),$$
where the constant $\mathcal{L}(d, p, q)$ is defined for $0 < p < q \le \infty$ by
$$\mathcal{L}(d, p, q) := \sup \{\|{f}\|_{L^q(\mathbb{R}^d)}: f \in \mathcal{E}_p^d, \|{f}\|_{L^p(\mathbb{R}^d)} = 1\}$$
with $\mathcal{E}_p^d$ denoting the set of all entire functions $f \in L^p(\mathbb{R}^d)$ of spherical exponential type
at most $1$.
These results extend the recent results of Levin and Lubinsky for trigonometric polynomials
on the unit circle.
Compared with those in one variable, our proof in higher-dimensional case is more
difficult because functions on the sphere can not be identified as periodic functions on
Euclidean space and explicit connections between spherical polynomial interpolation and
the Shannon sampling theorem for entire functions of exponential type are not available.
Our proof of the upper estimate relies on a recent deep result of Bondarenko, Radchenko
and Viazovska on spherical designs:
$$\frac{1}{|\mathbb{S}^d|}
\int\limits_{\mathbb{S}^d}
f(x) dx =
\frac{1}{N}
\sum\limits^{N}_{j=1}
f(x_{n,j} ),\quad f \in \Pi_n^d,$$
an earlier result of Yudin on the distribution of points of spherical designs $\{x_{n,j}\}$, and
also our previous result on a connection between positive cubature formulas and the
Marcinkiewitcz–Zygmund inequality on the sphere:
$$\|{f}\|_p \asymp
\left(\sum\limits_{\omega\in\Lambda}
\lambda_{n,j} |f(x_{n,j} )|^p
\right)^{1/p}
,\quad 0 < p < \infty.$$
The proof of the lower estimate is based on the de la Vallée-Poussin type kernels associated
with a smooth cutoff function on the sphere and also some properties of the exponential
mapping from the tangent plane to the sphere, which connects functions on sphere with
functions on Euclidean space.
2. While it remains a very challenging open problem to determine the exact value
of the Nikolskii constant $\mathcal{L}(d, 1, \infty)$, we are able to find the exact value of the Nikolskii
constant for $p = 1, q = \infty$ and nonnegative functions $f \in \mathcal{E}^d_1$:
$$\sup\limits_{0\le f\in\mathcal{E}^d_1}
\frac{\|{f}\|_{L^{\infty}(\mathbb{R}^d)}}{\|{f}\|_{L^{1}(\mathbb{R}^d)}}
=
\frac{1}{2^{d-1}|\mathbb{S}^d|\Gamma(d + 1)}.$$
3. We investigate the normalized Nikolskii constant
$$L_d :=\frac{|\mathbb{S}^d|\Gamma(d + 1)}{2}\mathcal{L}(d, 1,\infty).$$
For this problem, we first show existence of an extremal function and its uniqueness.
It was known that $L_d \le 1$ and $L_d \ge e^{-d(1+o(1))}$ as $d\to\infty$. We improve these bounds
as follows:
$$2^{-d} \le L_d \le {}_1F_2\left(
\frac{d}{2};\frac{d}{2}+1;\frac{d}{2}+1;-\frac{\beta^2_d}{4}\right),$$
where $_1F_2$ and $\beta_d$ denote the hypergeometric function and the smallest positive zero of the
Bessel function $J_{d/2}$, respectively. This implies that the constant $L_d$ decays exponentially
fast as $d \to\infty$:
$$(0.5)^d \le L_d \le (\sqrt{2/e})^{d(1+o(1))},\quad \sqrt{2/e} = 0.85776 \dots .$$
4. We observe that for $d \ge 2$, the asymptotic order of the usual Nikolskii inequality
on $\mathbb{S}^d$
can be signi cantly improved in many cases, for lacunary spherical polynomials of
the form $f =
\sum\limits^m_{j=0} f_{n_j}$ with $f_{n_j}$ being a spherical harmonic of degree $n_j$ and $n_{j+1}-n_j \ge 3$.
As is well known, for $d = 1$, the Nikolskii inequality for trigonometric polynomials on the
unit circle does not have such a phenomenon.
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