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The expected $L_{p}$ norm of random polynomials


Authors: Peter Borwein and Richard Lockhart
Journal: Proc. Amer. Math. Soc. 129 (2001), 1463-1472
MSC (1991): Primary 26D05
Published electronically: October 25, 2000
MathSciNet review: 1814174
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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

\begin{displaymath}q_{n}(\theta ) = \sum_{0}^{n-1}X_{k}e^{ik\theta}\end{displaymath}

let


\begin{displaymath}\vert\vert q_{n}\vert\vert _{p}^{p}= \int _{0}^{2\pi } \vert q_{n}(\theta )\vert^{p} \,d\theta /(2\pi ). \end{displaymath}

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 ${\mathrm E}(\vert X_{k}\vert^{p})$. Then


\begin{displaymath}\frac{\text{ E}(\vert\vert q_{n}\vert\vert _{p}^{p})}{n^{p/2}} \to \Gamma (1+p/2) \end{displaymath}

and


\begin{displaymath}\frac{\text{E}(\vert\vert q_{n}^{(r)}\vert\vert _{p}^{p})}{n^{(2r+ 1)p/2}} \to (2r+1)^{-p/2}\Gamma (1+p/2) \end{displaymath}

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


\begin{displaymath}\frac{\text{ E}(\vert\vert q_{n}\vert\vert _{p})}{n^{1/2}} \to (\Gamma (1+p/2))^{1/p} \end{displaymath}

and

\begin{displaymath}\frac{\text{ E}(\vert\vert q_{n}^{(r)}\vert\vert _{p})}{n^{(2r+1)/2}} \to (2r+1)^{-1/2}(\Gamma (1+p/2))^{1/p}. \end{displaymath}

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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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-00-05690-2
Keywords: Random polynomial, Littlewood polynomial, expected $L_{p}$ norm
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
Article copyright: © Copyright 2000 by the authors