Average Nikolskii factors for random spherical polynomials

For 1 ≤ p , q ≤ ∞ 1≤ p,q≤ ∞ , the Nikolskii factor for a spherical polynomial P a P_{{ a}} is defined by N p , q ( P a ) = ‖ P a ‖ q ‖ P a ‖ p , P a ( x ) = ∑ k = 1 N a k φ k ( x ) , {equation*} Np,q(P_{{ a}})= {\|P_{{ a}}\|q}{\|P_{{ a}}\|ₚ},\ \ P_{{ a}}(x)=∑ Nₖ₌₁aₖφ ₖ(x), {equation*} where { φ k } k = 1 N \{φ _k\}ₖ₌₁^N is an orthonormal basis in the space Π n d Π _n^d of all spherical polynomials of degree at most n n on S d − 1 Sᵈ⁻¹ , and N = dim ⁡ Π n d N= Π _n^d . We study the average Nikolskii factor for random spherical polynomials with independent N ( 0 , σ 2 ) N(0,σ ²) coefficients and obtain the exact order. For 1 ≤ p > q > ∞ 1≤ p>q>∞ , the average Nikolskii factor is of order N 0 N⁰ (i.e., constant), as compared to the worst case bound of order N 1 / p − 1 / q N1/p-1/q , and for 1 ≤ p > q = ∞ 1≤ p>q=∞ , the average Nikolskii factor is of order ( ln ⁡ N ) 1 / 2 (ln N)1/2 as compared to the worst case bound of order N 1 / p N1/p . We also give the generalization to the compact two-point homogeneous spaces.