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},
  journal={Canadian Journal of Mathematics},
  year={2016},
  volume={69},
  pages={807 - 825}
}
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… 

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References

SHOWING 1-10 OF 43 REFERENCES

Littlewood Polynomials with Small $L^4$ Norm

Asymptotic $L^4$ norm of polynomials derived from characters

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.

Some theorems on Fourier coefficients

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

An extremal property of Fekete polynomials

. 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

SOME NEW RESULTS ON THE RUDIN-SHAPIRO POLYNOMIALS

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

Finite Fields

  • K. Conrad
  • Mathematics
    Series and Products in the Development of Mathematics
  • 2004
This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields. We write Z/(p) and Fp interchangeably for the

A sequence of integers related to the Bessel functions

(2»)! where B» = (B + 1)» (n * 1), Gn = 2(1 2")B„. In view of the known arithmetic properties of these and related numbers, it is of some interest to look for arithmetic properties of <rtn(v) for

Computational Excursions in Analysis and Number Theory

Preface.- Introduction.- LLL and PSLQ.- Pisot and Salem Numbers.- Rudin-Shapiro Polynomials.- Fekete Polynomials.- Products of Cyclotomic Polynomials.- Location of Zeros.- Maximal Vanishing.-

Gauss Sums, Kloosterman Sums, And Monodromy Groups

The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's

Advances in the merit factor problem for binary sequences