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Mathematics of Computation

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Rudin-Shapiro-like polynomials in $L_{4}$

Authors: Peter Borwein and Michael Mossinghoff
Journal: Math. Comp. 69 (2000), 1157-1166
MSC (1991): Primary 11J54, 11B83, 12-04
Published electronically: March 2, 2000
MathSciNet review: 1709147
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Abstract: We examine sequences of polynomials with $\{+1,-1\}$ coefficients constructed using the iterations $p(x)\rightarrow p(x)\pm x^{d+ 1}p^{*}(-x)$, where $d$ is the degree of $p$ and $p^{*}$ is the reciprocal polynomial of $p$. If $p_{0}=1$ these generate the Rudin-Shapiro polynomials. We show that the $L_{4}$ norm of these polynomials is explicitly computable. We are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic $L_{4}$ norm (or, equivalently, with largest possible asymptotic merit factor). The Rudin-Shapiro polynomials form one such sequence.

We determine all $p_{0}$ of degree less than 40 that generate sequences under the iteration with this property. These sequences have asymptotic merit factor 3. The first really distinct example has a $p_{0}$ of degree 19.

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Additional Information

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

Michael Mossinghoff
Affiliation: Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608
Address at time of publication: Department of Mathematics, UCLA, Los Angeles, California 90095

Keywords: Restricted coefficients; $-1,0,1$ coefficients; Rudin-Shapiro polynomials; Littlewood conjectures
Received by editor(s): April 14, 1998
Published electronically: March 2, 2000