Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 242-255
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M. A. Shagay; N. A. Shirokov. Polynomial approximation by doubly periodic Weierstrass functions on disjoint segments in the $L^P$ metric. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 242-255. http://geodesic.mathdoc.fr/item/ZNSL_2023_527_a10/
@article{ZNSL_2023_527_a10,
author = {M. A. Shagay and N. A. Shirokov},
title = {Polynomial approximation by doubly periodic {Weierstrass} functions on disjoint segments in the $L^P$ metric},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {242--255},
year = {2023},
volume = {527},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_527_a10/}
}
TY - JOUR
AU - M. A. Shagay
AU - N. A. Shirokov
TI - Polynomial approximation by doubly periodic Weierstrass functions on disjoint segments in the $L^P$ metric
JO - Zapiski Nauchnykh Seminarov POMI
PY - 2023
SP - 242
EP - 255
VL - 527
UR - http://geodesic.mathdoc.fr/item/ZNSL_2023_527_a10/
LA - ru
ID - ZNSL_2023_527_a10
ER -
%0 Journal Article
%A M. A. Shagay
%A N. A. Shirokov
%T Polynomial approximation by doubly periodic Weierstrass functions on disjoint segments in the $L^P$ metric
%J Zapiski Nauchnykh Seminarov POMI
%D 2023
%P 242-255
%V 527
%U http://geodesic.mathdoc.fr/item/ZNSL_2023_527_a10/
%G ru
%F ZNSL_2023_527_a10
Let $s_k, 1\leqslant k\leqslant m, m\geqslant 2$, be disjoint segments lying in a parallelogram $Q$. We denote by $\wp(z)$ a doubly periodic Weierstrass function with the fundamental parallelogram $Q$. Let $f_k:s_k\rightarrow\mathbb{C}$ be functions, and let $f_k'\in L^{p_k}(s_k), 1\leqslant k\leqslant m, 1. Consider the Green function $G(z)$ of the domain $\mathbb{C}\backslash\overset{m}{\underset{k=1}{\cup}} s_k$ with the pole at infinity and define $$ L_h\stackrel{\rm def}{=} \{\ \zeta: \zeta\in\mathbb{C}\backslash\overset{m}{\underset{k=1}{\cup}} s_k, G(\zeta)=\log(1+h) \}, h>0; \rho_h(\zeta)\stackrel{\rm def}{=} \mathrm{dist}(\zeta,L_h). $$ Theorem. There exist polynomials $P_n(u,v), \deg P_n\leqslant n, n=1,2,\cdots$, such that $$ \overset{m}{\underset{k=1}{\sum}}{\underset{s_k}{\int}}\left|\frac{f_k(\zeta)-P_n(\wp(\zeta),\wp'(\zeta))}{\rho_{\frac1n}(\zeta)}\right|^{p_k}|d\zeta|\leqslant c. $$
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