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 ${\int }_{{\mathbb{S}}^d}d\sigma \left(x\right)=1$. This paper establishes a close connection between the asymptotic Nikolskii constant, ${\mathcal{L}}^{\ast }\left(d\right)≔\underset{n\to \infty }{lim}\frac{1}{dim{\Pi }_n^d}\underset{f\in {\Pi }_n^d}{sup}\frac{\Vert f{\Vert }_{L^{\infty }\left({\mathbb{S}}^d\right)}}{\Vert f{\Vert }_{L^1\left({\mathbb{S}}^d\right)}},$ and the following extremal problem: ${\mathcal{I}}_{\alpha }≔\underset{a_k}{inf}\Vert j_{\alpha +1}\left(t\right)-\sum _{k=1}^{\infty }{a}_k{j}_{\alpha }{\left(q_{\alpha +1,k}t∕q_{\alpha +1,1}\right)\Vert }_{L^{\infty }\left({\mathbb{R}}_{+}\right)}$ with the infimum being taken over all sequences ${\left\{a_k\right\}}_{k=1}^{\infty }\subset \mathbb{R}$ 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 $j_{\alpha }\left(0\right)=1$, and ${\left\{q_{\alpha +1,k}\right\}}_{k=1}^{\infty }$ denotes the strict increasing sequence of all positive zeros of $j_{\alpha +1}$. We prove that for $\alpha \ge -0.272$, ${\mathcal{I}}_{\alpha }=\frac{{\int }_0^{q_{\alpha +1,1}}{j}_{\alpha +1}\left(t\right)t^{2\alpha +1}dt}{{\int }_0^{q_{\alpha +1,1}}{t}^{2\alpha +1}dt}{=}_1{F}_2\left(\alpha +1;\alpha +2,\alpha +2;-\frac{q_{\alpha +1,1}^2}{4}\right).$ As a result, we deduce that the constant ${\mathcal{L}}^{\ast }\left(d\right)$ goes to zero exponentially fast as $d\to \infty$: $0.5^d\le {\mathcal{L}}^{\ast }\left(d\right)\le {(0.857\cdots )}^{d(1+{\epsilon }_d)}\text{with }{\epsilon }_d=O\left(d^{-2∕3}\right).$

让 ${\Pi }_{ñ}^d$ 最多表示度数的球面多项式的空间 $ñ$ 在单位范围内 ${\mathbb{小号}}^d\subset {\mathbb{[R}}^{d+\mathrm{1个}}$ 配备了表面勒贝格测度 $d\sigma$ 归一化 ${\int }_{{\mathbb{小号}}^d}d\sigma （X）=\mathrm{1个}$。本文建立了渐近的Nikolskii常数之间的紧密联系，${\mathcal{大号}}^{\ast }（d）≔\underset{ñ\to \infty }{林}\frac{\mathrm{1个}}{暗淡{\Pi }_{ñ}^d}\underset{F\in {\Pi }_{ñ}^d}{SUP}\frac{\Vert F{\Vert }_{{\mathrm{大号}}^{\infty }（{\mathbb{小号}}^d）}}{\Vert F{\Vert }_{{\mathrm{大号}}^{\mathrm{1个}}（{\mathbb{小号}}^d）}}，$ 以及以下极端问题： ${\mathcal{一世}}_{\alpha }≔\underset{{\mathrm{一种}}_{ķ}}{信息}\Vert {Ĵ}_{\alpha +\mathrm{1个}}（Ť）-\sum _{ķ=\mathrm{1个}}^{\infty }{\mathrm{一种}}_{ķ}{Ĵ}_{\alpha }{（q_{\alpha +\mathrm{1个}，ķ}Ť∕q_{\alpha +\mathrm{1个}，\mathrm{1个}}）\Vert }_{{\mathrm{大号}}^{\infty }（{\mathbb{[R}}_{+}）}$ 将所有序列取下 ${\left\{{\mathrm{一种}}_{ķ}\right\}}_{ķ=\mathrm{1个}}^{\infty }\subset \mathbb{[R}$ 使得无穷级数绝对收敛于 ${\mathbb{[R}}_{+}$。这里${Ĵ}_{\alpha }$ 表示第一种标准化的贝塞尔函数，因此 ${Ĵ}_{\alpha }（0）=\mathrm{1个}$， 和 ${\left\{q_{\alpha +\mathrm{1个}，ķ}\right\}}_{ķ=\mathrm{1个}}^{\infty }$ 表示的所有正零的严格递增顺序 ${Ĵ}_{\alpha +\mathrm{1个}}$。我们证明$\alpha \ge -0。272$， ${\mathcal{一世}}_{\alpha }=\frac{{\int }_0^{q_{\alpha +\mathrm{1个}，\mathrm{1个}}}{Ĵ}_{\alpha +\mathrm{1个}}（Ť）{Ť}^{\mathrm{2个}\alpha +\mathrm{1个}}dŤ}{{\int }_0^{q_{\alpha +\mathrm{1个}，\mathrm{1个}}}{Ť}^{\mathrm{2个}\alpha +\mathrm{1个}}dŤ}{=}_{\mathrm{1个}}{F}_{\mathrm{2个}}（\alpha +\mathrm{1个};\alpha +\mathrm{2个}，\alpha +\mathrm{2个};-\frac{q_{\alpha +\mathrm{1个}，\mathrm{1个}}^{\mathrm{2个}}}{4}）。$ 结果，我们推断出常数 ${\mathcal{大号}}^{\ast }（d）$ 随着 $d\to \infty$： $0。5^d\le {\mathcal{大号}}^{\ast }（d）\le {（0。857\cdots ）}^{d（\mathrm{1个}+{\epsilon }_d）}\text{和 }{\epsilon }_d=Ø（d^{-\mathrm{2个}∕3}）。$