Polynomial Approximation by Doubly Periodic Weierstrass Functions on Disjoint Segments in the L^P Metric

Let s_k, 1 ≤ k ≤ m, m ≥ 2, be disjoint segments lying in a parallelogram Q. Denote by ℘(z) a doubly periodic Weierstrass function with the fundamental parallelogram Q. Let f_k : s_k → ℂ be functions, and let \({f}_{k}^{\prime}\) $f_k^{\prime }$ ∈ \({L}^{{p}_{k}}\) $L^{p_k}$ (s_k), 1 ≤ k ≤ m, 1 < p_k < ∞.

Consider the Green function G(z) of the domain \({\mathbb{C}}\backslash \bigcup_{k=1}^{m}{s}_{k}\) $\mathbb{C}\backslash {\bigcup }_{k=1}^m{s}_k$ with the pole at infinity and define \(\begin{array}{ccc}{L}_{h}\stackrel{\scriptscriptstyle\mathrm{def}}{=}\left\{\zeta :\zeta \in {\mathbb{C}}\backslash \bigcup\limits_{k=1}^{m}{s}_{k},G\left(\zeta \right)=\mathrm{log}\left(1+h\right)\right\},\ \ h>0;\ \ {\rho }_{h}\left(\zeta\right)\stackrel{\scriptscriptstyle\mathrm{def}}{=}\mathrm{dist}\left(\zeta ,{L}_{h}\right)\end{array}.\) $$\begin{array}{c}{L}_h\stackrel{\mathrm{def}}{=}\left\{\zeta :\zeta \in \mathbb{C}\backslash ,\bigcup _{k=1}^m,s_k,,,G,\left(\zeta \right),=,\log ,\left(1,+,h\right)\right\},h>0;{\rho }_h\left(\zeta \right)\stackrel{\mathrm{def}}{=}\mathrm{dist}\left(\zeta ,,,L_h\right)\end{array}.$$

Theorem. There exist polynomials P_n(u, v), deg P_n ≤ n, n = 1, 2, · · · , such that \(\sum_{k=1}^{m}\underset{{s}_{k}}{\int }{\left|\frac{{f}_{k}\left(\zeta \right)-{P}_{n}\left(\mathrm{\wp }\left(\zeta \right),{\mathrm{\wp }}^{\mathrm{^{\prime}}}\left(\zeta \right)\right)}{{\rho }_\frac{1}{n}\left(\zeta \right)}\right|}^{{p}_{k}}\left|d\zeta \right|\le c.\) $$\sum _{k=1}^m\underset{s_k}{\int }\left|\frac{f_k\left(\zeta \right)-P_n\left(\wp ,\left(\zeta \right),,,\wp _{}^{_{}^{\prime }},\left(\zeta \right)\right)}{{\rho }_{\frac{1}{n}}\left(\zeta \right)}\right|_{}^{p_k}\left|d,\zeta \right|\le c.$$