Estimates of the asymptotic Nikolskii constants for spherical polynomials

Abstract

Let Πnd denote the space of spherical polynomials of degree at most n on the unit sphere Sd⊂Rd+1 that is equipped with the surface Lebesgue measure dσ normalized by ∫Sddσ(x)=1. This paper establishes a close connection between the asymptotic Nikolskii constant, L∗(d)≔limn→∞1dimΠndsupf∈Πnd‖f‖L∞(Sd)‖f‖L1(Sd), and the following extremal problem: Iα≔infak‖jα+1(t)−∑k=1∞akjα(qα+1,kt∕qα+1,1)‖L∞(R+) with the infimum being taken over all sequences {ak}k=1∞⊂R such that the infinite series converges absolutely a.e. on R+. Here jα denotes the Bessel function of the first kind normalized so that jα(0)=1, and {qα+1,k}k=1∞ denotes the strict increasing sequence of all positive zeros of jα+1. We prove that for α≥−0.272, Iα=∫0qα+1,1jα+1(t)t2α+1dt∫0qα+1,1t2α+1dt=1F2(α+1;α+2,α+2;−qα+1,124). As a result, we deduce that the constant L∗(d) goes to zero exponentially fast as d→∞: 0.5d≤L∗(d)≤(0.857⋯)d(1+εd)with εd=O(d−2∕3).