Article
Ufa Mathematical Journal
Volume 14, Number 3, pp. 117-126
Maximal convergence of Faber series in weighted rearrangement invariant Smirnov classes
Testici A.
DOI:10.13108/2022-14-3-117
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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 $G_{R}:=\text{Int}\,\Gamma _{R}$. 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.