Polynomial approximation by doubly periodic Weierstrass functions on disjoint segments in the $L^P$ metric

M. A. Shagay ; N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 242-255

Résumé

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 $$ \overset{m}{\underset{k=1}{\sum}}{\underset{s_k}{\int}}\left|\frac{f_k(\zeta)-P_n(\wp(\zeta),\wp'(\zeta))}{\rho_{\frac1n}(\zeta)}\right|^{p_k}|d\zeta|\leqslant c. $$