# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## The expected $L_{p}$ norm of random polynomialsHTML articles powered by AMS MathViewer

by Peter Borwein and Richard Lockhart
Proc. Amer. Math. Soc. 129 (2001), 1463-1472

## Abstract:

The results of this paper concern the expected $L_{p}$ norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval $[0, 2\pi ]$). Specifically, for a random polynomial $q_{n}(\theta ) = \sum _{0}^{n-1}X_{k}e^{ik\theta }$ let $||q_{n}||_{p}^{p}= \int _{0}^{2\pi } |q_{n}(\theta )|^{p} d\theta /(2\pi ).$ Assume the random variables $X_{k},k\ge 0$, are independent and identically distributed, have mean 0, variance equal to 1 and, if $p>2$, a finite $p^{th}$ moment ${\mathrm E}(|X_{k}|^{p})$. Then $\frac {\text { E}(||q_{n}||_{p}^{p})}{n^{p/2}} \to \Gamma (1+p/2)$ and $\frac {\text {E}(||q_{n}^{(r)}||_{p}^{p})}{n^{(2r+ 1)p/2}} \to (2r+1)^{-p/2}\Gamma (1+p/2)$ as $n\to \infty$. In particular if the polynomials in question have coefficients in the set $\{+1,-1\}$ (a much studied class of polynomials), then we can compute the expected $L_{p}$ norms of the polynomials and their derivatives $\frac {\text { E}(||q_{n}||_{p})}{n^{1/2}} \to (\Gamma (1+p/2))^{1/p}$ and $\frac {\text { E}(||q_{n}^{(r)}||_{p})}{n^{(2r+1)/2}} \to (2r+1)^{-1/2}(\Gamma (1+p/2))^{1/p}.$ This complements results of Fielding in the $p:=0$ case, Newman and Byrnes in the $p:=4$ case, and Littlewood et al. in the $p=\infty$ case.
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• Peter Borwein
• Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
• Email: pborwein@cecm.sfu.ca
• Richard Lockhart
• Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
• Email: lockhart@sfu.ca
• Received by editor(s): December 18, 1998
• Received by editor(s) in revised form: August 31, 1999
• Published electronically: October 25, 2000
• Additional Notes: Research supported in part by the NSERC of Canada.
• Communicated by: Albert Baernstein II
• © Copyright 2000 by the authors
• Journal: Proc. Amer. Math. Soc. 129 (2001), 1463-1472
• MSC (1991): Primary 26D05
• DOI: https://doi.org/10.1090/S0002-9939-00-05690-2
• MathSciNet review: 1814174