THE MAXIMAL VALUE OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS

Title & Authors
THE MAXIMAL VALUE OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS
Dubicks, Arturas; Jankauskas, Jonas;

Abstract
Let $\small{\zeta}$ be a fixed complex number. In this paper, we study the quantity $\small{S(\zeta,\;n):=mas_{f{\in}{\Lambda}_n}\;|f(\zeta)|}$, where $\small{{\Lambda}_n}$ is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given $\small{{\zeta}\;{\in}\;{\mathbb{C}}}$ and $\small{n\;{\in}\;{\mathbb{N}}}$, the quantity S($\small{\zeta}$, n) can be calculated. Then we compute the limit $\small{lim_{n{\rightarrow}{\infty}}\;S(\zeta,\;n)/n}$ for every $\small{{\zeta}\;{\in}\;{\mathbb{C}}}$ of modulus 1. It is equal to 1/$\small{\pi}$ if $\small{\zeta}$ is not a root of unity. If $\small{\zeta\;=\;\exp(2{\pi}ik/d)}$, where $\small{d\;{\in}\;{\mathbb{N}}}$ and k $\small{\in}$ [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin($\small{\pi}$/d)) and 1/(2d sin($\small{\pi}$/2d)) for d = 1, d even and d > 1 odd, respectively.
Keywords
Newman polynomial;maximum of a polynomial;root of unity;Dirichlet's theorem;
Language
English
Cited by
1.
Asymptotic results for a class of triangular arrays of multivariate random variables with Bernoulli distributed components∗, Lithuanian Mathematical Journal, 2016, 56, 3, 298
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