Estimates of the asymptotic Nikolskii constants for spherical polynomials

Let ${\mathrm{\Pi }}_n^d$ 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, $${\mathcal{L}}^{\ast }(d):=\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{\dim {\mathrm{\Pi }}_n^d}\underset{f\in {\mathrm{\Pi }}_n^d}{\sup }\frac{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{S}}^d)}}{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1({\mathbb{S}}^d)}},$$ and the following extremal problem: $${\mathcal{I}}_{\alpha }:=\underset{a_k}{\inf }\parallel j_{\alpha +1}(t)-\sum _{k=1}^{\mathrm{\infty }}{a}_k{j}_{\alpha }\left(q_{\alpha +1,k}t\mathrm{/}{q}_{\alpha +1,1}\right){\parallel }_{L^{\mathrm{\infty }}({\mathbb{R}}_{+})}$$ with the infimum being taken over all sequences ${a_k}{k=1}^\infty\subset $\mathbb{R}$$suchthattheinfiniteseriesconvergesabsolutelya\mathrm{.}e\mathrm{.}on$$\mathbb{R}$+$\mathrm{.}Here$j_\alpha $ denotes the Bessel function of the first kind normalized so that $j_{\alpha }(0)=1$, and $\{q_{\alpha +1,k}{\}}_{k=1}^{\mathrm{\infty }}$ denotes the strict increasing sequence of all positive zeros of $j_{\alpha +1}$. We prove that for $\alpha \ge -0.272$, $${\mathcal{I}}_{\alpha }=\frac{{\int }_0^{q_{\alpha +1,1}}{j}_{\alpha +1}(t)t^{2\alpha +1}dt}{{\int }_0^{q_{\alpha +1,1}}{t}^{2\alpha +1}dt}={}_1{F}_2(\alpha +1;\alpha +2,\alpha +2;-\frac{q_{\alpha +1,1}^2}{4})\mathrm{.}$$ As a result, we deduce that the constant ${\mathcal{L}}^{\ast }(d)$ goes to zero exponentially fast as $d\to \mathrm{\infty }$: $${0.5}^d\le {\mathcal{L}}^{\ast }(d)\le (0.857\cdots {)}^{d(1+{\epsilon }_d)}\text{with }{\epsilon }_d=O(d^{-2\mathrm{/}3})\text{.}$$