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- W4297919697 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, $0<pleinfty,$ 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 $Vert P_n' Vert/Vert P_nVert$ is often different from the classical deterministic upper bound $n$. In particular, for circles of radii less than one, the ratio $Vert P_n' Vert/Vert P_nVert$ 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 $Vert P_n' Vert/Vert P_nVert$ 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 $Vert P_n' Vert/Vert P_nVert$ 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." @default.
- W4297919697 created "2022-10-01" @default.
- W4297919697 creator A5067786184 @default.
- W4297919697 creator A5061523923 @default.
- W4297919697 date "2018-10-23" @default.
- W4297919697 modified "2023-09-27" @default.
- W4297919697 title "Random Bernstein-Markov factors" @default.
- W4297919697 doi "https://doi.org/10.48550/arxiv.1810.09931" @default.
- W4297919697 hasPublicationYear "2018" @default.
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