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 $P_{\mathbf{a}}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\mathbf{a}})=\frac{\Vert P_{\mathbf{a}}{\Vert }_q}{\Vert P_{\mathbf{a}}{\Vert }_p},P_{\mathbf{a}}(\mathbf{x})=\sum _{k:\hat{I}{»}_k\le n}{a}_k\ddot{I}{•}_k(\mathbf{x}),$$ where $\mathbf{a}=\{a_k{\}}_{\hat{I}{»}_k\le n}$, and $\{(\ddot{I}{•}_k,-\hat{I}{»}_k^2){\}}_{k=0}^{\infty }$ are the eigenpairs of the Laplace-Beltrami operator $\hat{I}{”}_{\mathbb{M}}$ on a closed smooth Riemannian manifold $\mathbb{M}$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,\ddot{I}{ƒ}^2)$ coefficients and obtain the exact orders. For $1\le p<q<\infty$, 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 $1\le p<q=\infty$, the average Nikolskii factor is of order $(\ln n{)}^{1/2}$ as compared to the worst case bound of order $n^{d/p}$.

14 pages