On convergence in the mean (Q1918673)

Let \(G\) be an arbitrary bounded simply connected domain in \(\mathbb{C}\) with a rectifiable Jordan boundary \(\Gamma\). A function \(f\) holomorphic on \(G\) is said to be of the class \(E_p\) if there exists a constant \(K>0\) such that \(\int_{\Gamma_r} |f(z)|^p |dz|\leq K\) for any closed, rectifiable curve \(\Gamma_r \subset G\). It is shown that for domains \(G\) with a special boundary curve \(\Gamma\) the functions \(f\) of class \(E_p\) can be expanded into series of a certain Walsh system \((M_k)_k\) of rational functions with given sets of poles with convergence in the mean: \[ \lim_{n\to\infty} \int_\Gamma \Bigl|f(z)-\sum^n_{k=1} a_kM_k(z) \Bigr|^p |dz |=0. \]