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This article is cited in 3 scientific papers (total in 3 papers)
On estimates for Goncharov polynomials
V. A. Oskolkov
Abstract:
The following is proved in the article.
Theorem. {\it If a sequence of interpolation points satisfies the conditions $|\arg z_n|\leqslant\frac\pi2\left(1-\frac1\rho\right)$ for all sufficiently large $n$ and $\varlimsup_{n\to\infty}n^{-1/\rho}|z_n|=\varlimsup_{n\to\infty}n^{-1/\rho}S_n=1,$ where $S_n=\sum_{\nu=0}^{n-1}|z_\nu-z_{\nu+1}|,$ for $1\leqslant\rho<\infty,$ and $\arg z_n=0,$ $z_n\leqslant z_{n+1}$ $(n=0,1,\dots),$ $\lim_{n\to\infty}n^{-1/\rho}z_n=1$ for $0<\rho<1,$ then the assertions}
1) $\varlimsup_{n\to\infty}\{n^{-n/\rho}n!\max_{|z|\leqslant r}|P_n(z)|\}^{1/n}\equiv1$ for $1\leqslant\rho<\infty$,
2) $\frac1\rho\exp\left(1-\frac1\rho\right)\leqslant\varlimsup_{n\to\infty}\{n^{-n/\rho}n!\max_{|z|\leqslant r}|P_n(z)|\}^{1/n}\leqslant1$ for $0<\rho<1$
\noindentare valid for any $r<\infty$. Here $P_n(z)$ is the Goncharov polynomial of degree $n$.
Bibliography: 3 titles.
Received: 23.10.1972
Citation:
V. A. Oskolkov, “On estimates for Goncharov polynomials”, Mat. Sb. (N.S.), 92(134):1(9) (1973), 55–59; Math. USSR-Sb., 21:1 (1973), 57–62
Linking options:
https://www.mathnet.ru/eng/sm3331https://doi.org/10.1070/SM1973v021n01ABEH002005 https://www.mathnet.ru/eng/sm/v134/i1/p55
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Abstract page: | 262 | Russian version PDF: | 75 | English version PDF: | 3 | References: | 28 |
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