# Lq Norms of Fekete and Related Polynomials

@article{Gnther2016LqNO,
title={Lq Norms of Fekete and Related Polynomials},
author={Christian G{\"u}nther and Kai-Uwe Schmidt},
year={2016},
volume={69},
pages={807 - 825}
}
• Published 4 February 2016
• Mathematics
Abstract A Littlewood polynomial is a polynomial in $\mathbb{C}\left[ z \right]$ 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 ${{L}^{q}}$ normon the complex unit circle. We consider the Fekete polynomials $${{f}_{p}}(z)=\sum\limits_{j=1}^{p-1}{(j|p){{z}^{j… 6 Citations This work shows that there is an absolute constant c > 1/2 such that the Mahler measure of the Fekete polynomials f_p, for all sufficiently large primes p, is at least c\sqrt{p}$$cp forall sufficiently largePrimes p.
• Mathematics
Mathematika
• 2022
For X(n)$X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials PN(θ)=1N∑n⩽NX(n)e(nθ),\begin{equation*} \hspace*{13pc}P_N(\theta ) = \frac{1}{\sqrt {N}}
• T. Erdélyi
• Materials Science
Constructive Approximation
• 2017
We show that there is an absolute constant c>1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}
• T. Erdélyi
• Mathematics
Trigonometric Sums and Their Applications
• 2020
This survey gives a taste of the author’s recent work on polynomials with constrained coefficients. Special attention is paid to unimodular, Littlewood, Newman, Rudin-Shapiro, and Fekete polynomials,
It is proved that the Rudin-Shapiro polynomials are not $L^\alpha$-flat, for any $\alpha \geq 0$.

## References

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This is the first time that the lowest known asymptotic ratio of norms for multivariable polynomials $f(z_1,...,z_n)$ is strictly less than what could be obtained by using products of the best known univariate polynmials.
where P is given by (1.1)? If one allows the coefficients Cn to be complex numbers of absolute value 1, an affirmative answer to the question is furnished by the partial sums of the series ZE
• Mathematics
• 2000
. The Fekete polynomials are de(cid:12)ned as where is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they
• Mathematics
• 2008
In this article, we focuss on. sequences of polynomials with {} coefficients constructed by recursive argument that is known as Rudin-Shapiro polynomials. The asymptotic behavior of these polynomials