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The expected norm of random polynomials

Authors:	Peter Borwein Richard Lockhart

Institution:	Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 ; Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

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 , 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 case, Newman and Byrnes in the case, and Littlewood et al. in the $p=\infty$ case.

Keywords:	Random polynomial Littlewood polynomial expected $L_{p}$ norm

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