For weighted $L_2$ spaces with doubling weights $w$ on $[-1,1]$ and norms ${\Vert \cdot \Vert }_{2,w}$, the Markov factor on a polynomial $P$ is defined by $\frac{{\Vert P^{\prime }\Vert }_{2,w}}{{\Vert P\Vert }_{2,w}}$. We study this Markov factor on random polynomials with independent $N(0,1)$ coefficients, and show that the upper bound of the average (expected) Markov factor is order degree to the $3∕2$, as compared to the degree squared worst case upper bound.

中文翻译：

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平均个案加权 ${\mathrm{大号}}_2$ 权重加倍的马尔可夫因子

对于加权 ${\mathrm{大号}}_2$ 权重加倍的空间 $w$ 上 $[-\mathrm{1个}，\mathrm{1个}]$ 和规范 ${\Vert \cdot \Vert }_{2，w}$，多项式上的马尔可夫因子 $P$ 由定义 $\frac{{\Vert P^{\prime }\Vert }_{2，w}}{{\Vert P\Vert }_{2，w}}$。我们用独立的随机多项式研究此马尔可夫因子$ñ（0，\mathrm{1个}）$ 系数，并表明平均（预期）马尔可夫因子的上限是 $3∕2$，与度数最坏情况上限的平方相比。