Estimates of the asymptotic Nikolskii constants for spherical polynomials

Abstract

Let ${\Pi }_n^d$ denote the space of spherical polynomials of degree at most $n$ on the unit sphere ${\mathbb{S}}^d\subset {\mathbb{R}}^{d+1}$ that is equipped with the surface Lebesgue measure $d\sigma$ normalized by ${\int }_{{\mathbb{S}}^d}d\sigma (x)=1$. This paper establishes a close connection between the asymptotic Nikolskii constant, $${\mathcal{L}}^{\ast }(d):=\underset{n\to \infty }{\lim }\frac{1}{\dim {\Pi }_n^d}\underset{f\in {\Pi }_n^d}{\sup }\frac{\Vert f{\Vert }_{L^{\infty }({\mathbb{S}}^d)}}{\Vert f{\Vert }_{L^1({\mathbb{S}}^d)}},$$ and the following extremal problem: $${\mathcal{I}}_{\alpha }:=\underset{a_k}{\inf }\Vert j_{\alpha +1}(t)-\sum _{k=1}^{\infty }{a}_k{j}_{\alpha }(q_{\alpha +1,k}t/q_{\alpha +1,1}){\Vert }_{L^{\infty }({\mathbb{R}}_{+})}$$ with the infimum being taken over all sequences $\{a_k{\}}_{k=1}^{\infty }\subset \mathbb{R}$ such that the infinite series converges absolutely a.e. on ${\mathbb{R}}_{+}$. Here $j_{\alpha }$ denotes the Bessel function of the first kind normalized so that $j_{\alpha }(0)=1$, and $\{q_{\alpha +1,k}{\}}_{k=1}^{\infty }$ denotes the strict increasing sequence of all positive zeros of $j_{\alpha +1}$. We prove that for $\alpha \ge -0.272$, $${\mathcal{I}}_{\alpha }=\frac{{\int }_0^{q_{\alpha +1,1}}{j}_{\alpha +1}(t)t^{2\alpha +1}dt}{{\int }_0^{q_{\alpha +1,1}}{t}^{2\alpha +1}dt}={}_1{F}_2(\alpha +1;\alpha +2,\alpha +2;-\frac{q_{\alpha +1,1}^2}{4}).$$ As a result, we deduce that the constant ${\mathcal{L}}^{\ast }(d)$ goes to zero exponentially fast as $d\to \infty$: \[ 0.5^d\le \mathcal{L}^{*}(d)\le (0.857\cdots)^{d\,(1+\varepsilon_d)} \ \ \ \ \ \text{with ${\epsilon }_d=O(d^{-2/3})$.} \]

27 pages

References in corpus (1)

• On the large argument asymptotics of the Lommel function via Stieltjes transforms