Inverse theorem of polynomial approximation on an elliptic curve

M. A. Shagay

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 52, Tome 537 (2024), pp. 151-177

Résumé

Let $\wp(z)$ be a doubly periodic Weierstrass function with periods $2\omega_1$, $2\omega_2$, let $Q$ be a parallelogram with vertexes $0, 2\omega_1, 2\omega_2, 2(\omega_1+\omega_2)$, and let $s_k$, $1\leq k\leq m$, be pairwise disjoint segments, $s_k=[a_k,b_k]\subset Q$, $1\leq k\leq m$. We choose numbers $\varepsilon_{kn}>0$ satisfying the condition $\sum\limits_{k=1}^m \overset{\infty} {\underset{n=1}{\sum}}\varepsilon^2_{kn}\infty$. We denote by $g(z)$ the Green functions of the region $\mathbb{C}\setminus \bigcup\limits_{k=1}^{m} s_k$ with the logarithmic pole at $\infty$ and put $L_h=\{z\in Q\setminus \bigcup\limits_{k=1}^{m} s_k: g(z)=h\}$, $0$. Let $T(z)=(\wp(z),\wp'(z)), z\in Q$, \begin{equation*} d_{kn}(z)=1+\dfrac{1}{2^n\sqrt{\delta(T(z),T(a_k))\cdot\delta(T(z),T(b_k))}}, z\in s_k, \end{equation*} \begin{equation*} \delta((\zeta,w),(\zeta',w'))=\sqrt{|\zeta-\zeta'|^2+|w-w'|^2} .\end{equation*} We prove the following claim. Theorem 1$'$. Suppose that $2\leq p_k\infty, 1\leq k\leq m, f_k\in C(s_k)$. Assume that there exist polynomials $\mathsf{P}_{2^n}(u,v), \deg\mathsf{P}_{2^n}\leq 2^n$, and a constant $C_*$ such that for $n=1,2,...$ one has the estimate \begin{equation*} \sum\limits_{k=1}^m \int\limits_{s_k}\displaystyle\left|\frac{f_k(z) -\mathsf{P}_{2^n}(\wp(z),\wp'(z))}{\varepsilon_{kn}\rho_{2^{-n}}(z)}\right|^{p_k}d_{kn}(z)|dz|\leq C_{*}. \end{equation*} Then $f_k'(z)\in L^{p_k}(s_k), 1\leq k\leq m$.