Classes of $(\psi, \beta)$-differential functions of complex variable and approximation by linear averages of their Faber series

Abstract

We introduce the notion of $(\psi, \beta)$-derivative of a function of one complex variable, and define on the basis of this the classes $L_\beta ^{\psi}{\mathfrak N} (G)$  of $(\psi, \beta)$-differentiable analytic functions in a bounded domain $G$. The classes $L_\beta ^{\psi}{\mathfrak N} (G)$ consist of the Cauchy-type integrals whose densities $f(\zeta)$ are such that the induced functions $\tilde f(t)$  on the unit circle are periodic functions of classes $L_\beta ^{\psi}{\mathfrak N}$. We consider approximation of functions $f\in L_\beta ^{\psi}{\mathfrak N} (G)$  by algebraic polynomials constructed from their series expansions in Faber polynomials.