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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

The average norm of polynomials of fixed height


Authors: Peter Borwein and Kwok-Kwong Stephen Choi
Journal: Trans. Amer. Math. Soc. 359 (2007), 923-936
MSC (2000): Primary 11C08, 26C05
Published electronically: September 12, 2006
MathSciNet review: 2255202
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ n\ge 0$ be any integer and let

$\displaystyle \mathfrak{F}_n:=\left\{ \sum_{i=0}^na_iz^i : a_i = 0, \pm 1 \right\} $

be the set of all polynomials of height 1 and degree $ n$. Let

$\displaystyle \beta_n(m):=\frac{1}{3^{n+1}}\sum_{P\in \mathfrak{F}_n}\Vert P\Vert _m^m. $

Here $ \Vert P\Vert _m^m$ is the $ mth$ power of the $ L_m$ norm on the boundary of the unit disc. So $ \beta_n(m)$ is the average of the $ mth$ power of the $ L_m$ norm over $ \mathfrak{F}_n.$

In this paper we give exact formulae for $ \beta_n(m)$ for various values of $ m$. We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients $ \pm 1$ and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.



Theorem 0.1. For $ n\ge 0$, we have

$\displaystyle \beta_n(2)=\frac23 (n+1),$    
$\displaystyle \beta_n(4)=\frac89 n^2+\frac{14}{9}n+\frac23$    

and

$\displaystyle \beta_n(6)=\frac{16}{9}n^3+4n^2+\frac{26}{9}n+\frac23. $


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Kwok-Kwong Stephen Choi
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: kkchoi@cecm.sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03952-3
PII: S 0002-9947(06)03952-3
Keywords: Polynomials of height 1, Littlewood polynomials, average $L_p$ norm
Received by editor(s): June 19, 2001
Received by editor(s) in revised form: January 22, 2005
Published electronically: September 12, 2006
Additional Notes: The research of the first author was supported by MITACS and by NSERC of Canada, and the research of the second author was supported by NSERC of Canada.
Article copyright: © Copyright 2006 by the authors