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

   

 

Explicit merit factor formulae for Fekete and Turyn polynomials


Authors: Peter Borwein and Kwok-Kwong Stephen Choi
Journal: Trans. Amer. Math. Soc. 354 (2002), 219-234
MSC (1991): Primary 11J54, 11B83, 12-04
Published electronically: August 20, 2001
MathSciNet review: 1859033
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Abstract: We give explicit formulas for the $L_{4}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials

\begin{displaymath}f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac{k}{q}\right ) z^{k} \end{displaymath}

where $\left (\frac{\cdot }{q}\right )$ is the Legendre symbol. For example for $q$ an odd prime,

\begin{displaymath}\Vert f_{q}\Vert _{4}^{4} : = \frac{5q^{2}}{3}-3q+ \frac{4}{3} - 12 (h(-q))^{2} \end{displaymath}

where $h(-q)$ is the class number of $\mathbb{Q}(\sqrt {-q})$. Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of $f_{q}$. This is the sequence that has the largest known asymptotic merit factor. Explicitly,

\begin{displaymath}R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+[q/4] }{q}\right ) z^{k} \end{displaymath}

where $[\cdot ]$ denotes the nearest integer, satisfies

\begin{displaymath}\Vert R_{q}\Vert _{4}^{4} = \frac{7q^{2}}{6}- {q} - \frac{1}{6} - \gamma _{q} \end{displaymath}

where

\begin{displaymath}\gamma _{q}: = \begin{cases} h(-q) (h(-q)-4) & \text{if} \qu... ...pmod 8, \\ 0 & \text{if} \quad q \equiv 7 \pmod 8. \end{cases}\end{displaymath}

Indeed we derive a closed form for the $L_{4}$ norm of all shifted Fekete polynomials

\begin{displaymath}f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+t}{q}\right ) z^{k}. \end{displaymath}

Namely
\begin{align*}\Vert f_{q}^{t} \Vert _{4}^{4} &= \frac{1}{3}(5q^{2}+3q+4)+8t^{2}-... ...tyle \sum _{n=1}^{q-1}n\left(\frac{n+t}{q}\right)}\right \vert^{2}, \end{align*}
and $\Vert f_{q}^{q-t+1} \Vert _{4}^{4}= \Vert f_{q}^{t} \Vert _{4}^{4}$ if $1 \le t \le (q+1)/2$.


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6

Kwok-Kwong Stephen Choi
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02859-8
PII: S 0002-9947(01)02859-8
Keywords: Class number, $-1,1$ coefficients, merit factor, Fekete polynomials, Turyn polynomials, Littlewood polynomials
Received by editor(s): April 24, 2000
Published electronically: August 20, 2001
Additional Notes: Research of P. Borwein is supported, in part, by NSERC of Canada. K.K. Choi is a Pacific Institute of Mathematics Postdoctoral Fellow and the Institute’s support is gratefully acknowledged
Article copyright: © Copyright 2001 by the authors