Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds

Abstract

For $1\le p,q\le \infty$, the Nikolskii factor for a diffusion polynomial $P_{\bf a}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\bf a})=\frac{\|P_{\bf a}\|_{q}}{\|P_{\bf a}\|_{p}},\ \ P_{\bf a}({\bf x})=\sum_{k:\lambda_{k}\leq n}a_{k}\phi_{k}({\bf x}),$$ where ${\bf a}=\{a_k\}_{\lambda_k\le n}$, and $\{(\phi_k,-\lambda_k^2)\}_{k=0}^\infty$ are the eigenpairs of the Laplace-Beltrami operator $\Delta_{\mathbb M}$ on a closed smooth Riemannian manifold $\mathbb M$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,\sigma^{2})$ coefficients and obtain the exact orders. For $1\leq p<q<\infty$, the average Nikolskii factor is of order $n^{0}$ (i.e., constant), as compared to the worst case bound of order $n^{d(1/p-1/q)}$, and for $1\leq p<q=\infty$, the average Nikolskii factor is of order $(\ln n)^{1/2}$ as compared to the worst case bound of order $n^{d/p}$.