Polynomials least deviating from zero in $L^p(-1;1) $, $ 0 \le p \le \infty $, with a constraint on the location of
Ural mathematical journal, Tome 9 (2023) no. 2, pp. 157-164
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We study Chebyshev's problem on polynomials that deviate least from zero with respect to $L^p$-means on the interval $[-1;1]$ with a constraint on the location of roots of polynomials. More precisely, we consider the problem on the set $\mathcal{P}_n(D_R)$ of polynomials of degree $n$ that have unit leading coefficient and do not vanish in an open disk of radius $R \ge 1$. An exact solution is obtained for the geometric mean (for $p=0$) for all $R \ge 1$; and for $0$ for all $R \ge 1$ in the case of polynomials of even degree. For $0$ and $R\ge 1$, we obtain two-sided estimates of the value of the least deviation.
Keywords:
Algebraic polynomials, Chebyshev polynomials, сonstraints on the roots of a polynomial.
@article{UMJ_2023_9_2_a12,
author = {Alena E. Rokina},
title = {Polynomials least deviating from zero in $L^p(-1;1) $, $ 0 \le p \le \infty $, with a constraint on the location of},
journal = {Ural mathematical journal},
pages = {157--164},
publisher = {mathdoc},
volume = {9},
number = {2},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/UMJ_2023_9_2_a12/}
}
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Alena E. Rokina. Polynomials least deviating from zero in $L^p(-1;1) $, $ 0 \le p \le \infty $, with a constraint on the location of. Ural mathematical journal, Tome 9 (2023) no. 2, pp. 157-164. http://geodesic.mathdoc.fr/item/UMJ_2023_9_2_a12/