Maximal convergence of faber series in weighted rearrangement invariant smirnov classes

Abstract

Let K be a bounded set on the complex plane C with a connected complement (Formula Presented) and (Formula Presented) By φ we denote the conformal mapping of K−onto {w ∈ C: |w| > 1} normalized by the conditions φ (∞) = ∞ and (Formula Presented) and (Formula Presented). Let also (Formula Presented), k = 0, 1, 2,… be the Faber polynomials for K constructed via conformal mapping φ. As it is well known, if f is an analytic function in GR, then the representation (Formula Presented), z ∈ GR 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 EX (GR, ω) of analytic functions in GR is studied. Here the weight ω satisfies the Muckenhoupt condition on ΓR. The estimates are given in the uniform norm on K. The right sides of obtained inequalities involve the powers of the parameter R and En (f,G)X.ω called the best approximation number of f in EX (GR, ω), defined as En (Formula Presented). Here Πn 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.