Average Nikolskii factors for random trigonometric polynomials

Abstract

For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\mathbf{a}}$ is defined by $${\mathcal{N}}_{p,q}(T_{\mathbf{a}})=\frac{\Vert T_{\mathbf{a}}{\Vert }_q}{\Vert T_{\mathbf{a}}{\Vert }_p},T_{\mathbf{a}}(x)=a_1+\sum _{k=1}^n(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,\ddot{I}{ƒ}^2)$ coefficients and obtain that the exact order. For $1\le p<q<\infty$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1/p-1/q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials.

17 pages