Average Nikolskii factors for random trigonometric polynomials

Abstract

For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\bf a}$ is defined by $$\mathcal N_{p,q}(T_{\bf a})=\frac{\|T_{\bf a}\|_{q}}{\|T_{\bf a}\|_{p}},\ \ T_{\bf a}(x)=a_{1}+\sum\limits^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,\sigma^{2})$ coefficients and obtain that the exact order. For $1\leq p<q<\infty$, 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.