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$L^q$ Norms of Fekete and Related Polynomials

  Published:2016-06-28
 Printed: Aug 2017
  • Christian Günther,
    Department of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany
  • Kai-Uwe Schmidt,
    Department of Mathematics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany
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Abstract

A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.
Keywords: character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
MSC Classifications: 11B83, 42A05, 30C10 show english descriptions Special sequences and polynomials
Trigonometric polynomials, inequalities, extremal problems
Polynomials
11B83 - Special sequences and polynomials
42A05 - Trigonometric polynomials, inequalities, extremal problems
30C10 - Polynomials
 

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