Maximal convergence of Faber series in weighted rearrangement invariant Smirnov classes

Abstract

Let $K$ be a bounded set on the complex plane $\mathbb{C}$ with a connected complement $K^{-}:=\overline{\mathbb{C}}\backslash K$. Let $ \mathbb{D}:=\left\{ w\in \mathbb{C}:\left\vert w\right\vert <1\right\} $ and $\mathbb{D}^{-}:=\overline{\mathbb{C}}\backslash \overline{\mathbb{D}}$. By $\varphi $ we denote the conformal mapping of $K^{-}$onto $\left\{ w\in \mathbb{C} :\left\vert w\right\vert >1\right\} $ normalized by the conditions $\varphi \left( \infty \right) =\infty $ and $\lim_{z\rightarrow \infty}\varphi \left( z\right) /z>0$. Let $\Gamma _{R}:=\left\{ z\in K^{-}:\left\vert \varphi \left( z\right) \right\vert =R>1\right\} $ and $G_{R}:=\operatorname{Int}\Gamma _{R}$. Let also $\Phi _{k}\left( z\right) $, $k=0,1,2,\ldots$ be the Faber polynomials for $K$ constructed via conformal mapping $\varphi $. As it is well known, if $f $ is an analytic function in $G_{R}$, then the representation $ f\left( z\right) =\sum\limits_{k=0}^{\infty}a_{k}\left( f\right) \Phi _{k}\left( z\right) $, $z\in G_{R} $ holds. The partial sums of Faber series play an important role in constructing approximations in complex plane and investigating properties of Faber series is one of the essential issue. In this work the maximal convergence of the partial sums of the partial sums of the Faber series of $f$ in weighted rearrangement invariant Smirnov class $E_{X}\left(G_{R},\omega \right)$ of analytic functions in $G_{R}$ is studied. Here the weight $\omega$ satisfies the Muckenhoupt condition on $\Gamma _{R}$. The estimates are given in the uniform norm on $K$. The right sides of obtained inequalities involve the powers of the parameter $R$ and $E_{n}\left( f,G\right) _{X.\omega}$ called the best approximation number of $f$ in $E_{X}\left( G_{R},\omega \right) $, defined as $E_{n}\left( f,G\right) _{X.\omega}:=\inf \left\{ \left\Vert f-P_{n}\right\Vert _{X\left( \Gamma ,\omega \right)}:P_{n}\in \Pi _{n}\right\} $. Here $\Pi _{n}$ is the class of algebraic polynomials of degree not exceeding $n$. These results given in this paper is a kind of generalisation of similar results obtained in the classical Smirnov classes.