Random Bernstein-Markov factors

TL;DR

This paper investigates the behavior of Bernstein's inequality for random polynomials, revealing that the ratio of the norms of the derivative to the polynomial itself varies significantly from the classical bound, especially for different circle radii and norms.

Contribution

It provides new almost sure and expected bounds for the ratio of derivatives to polynomials in random settings, extending classical deterministic results.

Findings

For circles of radius less than one, the ratio is almost surely bounded as degree increases.

Expected ratio remains uniformly bounded under mild conditions on coefficients.

For radius greater than one, the ratio asymptotically equals n divided by the radius.

Abstract

For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials, and show that the expected (average) and almost sure value of $\Vert P_n' \Vert/\Vert P_n\Vert$ is often different from the classical deterministic upper bound $n$. In particular, for circles of radii less than one, the ratio $\Vert P_n' \Vert/\Vert P_n\Vert$ is almost surely bounded as $n$ tends to infinity, and its expected value is uniformly bounded for all degrees under mild assumptions on the random coefficients. For norms on the unit circle, Borwein and Lockhart mentioned that the asymptotic value of $\Vert P_n' \Vert/\Vert P_n\Vert$ in probability is $n/\sqrt{3},$ and we strengthen this to almost sure limit for $p=2.$ If the radius $R$ of the circle is larger than one, then the asymptotic value of $\Vert P_n' \Vert/\Vert P_n\Vert$ in probability is $n/R$, matching the sharp upper bound for the deterministic case. We also obtain bounds for the case $p=\infty$ on the unit circle.