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Let Π n d denote the space of spherical polynomials of degree at most n on the unit sphere S d ⊂ R d + 1 that is equipped with the surface Lebesgue measure d σ normalized by ∫ S d d σ (x) = 1. This paper establishes a close connection between the asymptotic Nikolskii constant, L ∗ (d) ≔ lim n → ∞ 1 dim Π n d sup f ∈ Π n d ‖ f ‖ L ∞ (S d) ‖ f ‖ L 1 (S d) , and the following extremal problem: I α ≔ inf a k ‖ j α + 1 (t) − ∑ k = 1 ∞ a k j α ( q α + 1 , k t ∕ q α + 1 , 1 ) ‖ L ∞ (R +) with the infimum being taken over all sequences { a k } k = 1 ∞ ⊂ R such that the infinite series converges absolutely a.e. on R +. Here j α denotes the Bessel function of the first kind normalized so that j α (0) = 1 , and { q α + 1 , k } k = 1 ∞ denotes the strict increasing sequence of all positive zeros of j α + 1. We prove that for α ≥ − 0. 272 , I α = ∫ 0 q α + 1 , 1 j α + 1 (t) t 2 α + 1 d t ∫ 0 q α + 1 , 1 t 2 α + 1 d t = 1 F 2 (α + 1 ; α + 2 , α + 2 ; − q α + 1 , 1 2 4 ). As a result, we deduce that the constant L ∗ (d) goes to zero exponentially fast as d → ∞ : 0. 5 d ≤ L ∗ (d) ≤ (0. 857 ⋯) d (1 + ε d) with ε d = O (d − 2 ∕ 3). [ABSTRACT FROM AUTHOR]