
Polynomials least deviating from zero on a square of the complex plane
E. B. Bayramov^{} ^{} Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
The Chebyshev problem is studied on the square $\Pi=ż=x+iy\in\mathbb{C}\colon\max\{x,y\}\le 1\}$ 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 twosided 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.
DOI:
https://doi.org/10.21538/013448892018243515
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UDC:
517.538+519.651
MSC: 30C10, 30C15, 30E10 Received: 01.07.2018
Citation:
E. B. Bayramov, “Polynomials least deviating from zero on a square of the complex plane”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 3, 2018, 5–15
Citation in format AMSBIB
\Bibitem{Bay18}
\by E.~B.~Bayramov
\paper Polynomials least deviating from zero on a square of the complex plane
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 3
\pages 515
\mathnet{http://mi.mathnet.ru/timm1545}
\crossref{https://doi.org/10.21538/013448892018243515}
\elib{http://elibrary.ru/item.asp?id=35511270}
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