E.B. Bairamov. Polynomials least deviating from zero on a square of the complex plane

The Chebyshev problem is studied on the square $\Pi=\left\{z=x+iy\in\mathbb{C}\colon\max\{|x|,|y|\}\le 1\right\}$ of the complex plane $\mathbb{C}$. Let $\mathfrak{P}_n$ be the set of algebraic polynomials of a given degree $n$ with the unit leading coefficient. The problem is to find the smallest value $\tau_n(\Pi)$ of the uniform norm $\|p_n\|_{C(\Pi)}$ of polynomials $p_n\in \mathfrak{P}_n$ on the square $\Pi$ and a polynomial with the smallest norm, which is called the Chebyshev polynomial (for the squire). The Chebyshev constant $\tau(Q)=\lim_{n\rightarrow\infty} \sqrt[n]{\tau_n(Q)}$ for the squire is found. Thus, the logarithmic asymptotics of the least deviation $\tau_n(\Pi)$ with respect to the degree of a polynomial is found. The problem is solved exactly for polynomials of degrees from 1 to 7. The class of polynomials in the problem is restricted; more exactly, it is proved that, for $n=4m+s$, $0\le s\le 3$, it is sufficient to solve the problem on the set of polynomials $z^sq_m(z)$, $q_m\in \mathfrak{P}_m$. Effective two-sided estimates for the value of the least deviation $\tau_n(\Pi)$ with respect to $n$ are obtained.

Keywords: algebraic polynomial, uniform norm, square of the complex plane, Chebyshev polynomial.

The paper was received by the Editorial Office on July 1, 2018.

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Emir Batyrovich Bairamov, Ural Federal University, Yekaterinburg, 620990 Russia,
e-mail: mrequ@yandex.ru


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