Polynomial approximations in a convex domain in $\mathbb C^n$ with the exponential decaying inside
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 49, Tome 503 (2021), pp. 154-171
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Let $\Omega$ be convex domain in $\mathbb C^n$ satisfying some restrictions, $f$ be holomorphic in $\Omega$ and continuons in $\overline{\Omega}$, $f\in H^{r+\omega}(\overline{\Omega})$ with a modulus of continuity $\omega$. Then there are polynomials $P_N$, $\deg P_N\le N$, such that $ |f(z)-P_N(z)| \le cN^{-r}\omega(\frac{1}{N})$, $z \in \overline{\Omega}$, and $|f(z)-P_N(z)| \le c \exp(-c_0(K)N)$, $z\in K\subset \Omega$, where $K$ is any compact strictly inside $\Omega$.
@article{ZNSL_2021_503_a9,
author = {N. A. Shirokov},
title = {Polynomial approximations in a convex domain in $\mathbb C^n$ with the exponential decaying inside},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {154--171},
year = {2021},
volume = {503},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_503_a9/}
}
N. A. Shirokov. Polynomial approximations in a convex domain in $\mathbb C^n$ with the exponential decaying inside. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 49, Tome 503 (2021), pp. 154-171. http://geodesic.mathdoc.fr/item/ZNSL_2021_503_a9/
[1] N. A. Shirokov, “Ravnomernye polinomialnye priblizheniya v vypuklykh oblastyakh v $\mathbb C^n$”, Zap. nauchn. semin. POMI, 333, 2006, 98–112 | Zbl
[2] L. A. Aizenberg, “Integralnoe predstavlenie funktsii, golomorfnykh v vypuklykh oblastyakh prostranstva $\mathbb C^n$”, DAN CCCR, 151:6 (1963), 1247–1249 | Zbl
[3] V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977
[4] N. A. Shirokov, “Polinomialnye $D$-priblizheniya, eksponentsialno ubyvayuschie strogo vnutri kontinuuma”, Algebra i analiz, 6:6 (1994), 146–176
[5] U. Rudin, Teoriya funktsii v edinichnom share iz $\mathbb C^n$, Mir, M., 1984