Classes of (ψ, β)-differentiable functions of a complex variable and approximation by linear averages of their Faber series

We introduce the notion of (ψ, Β)-derivative of a function of one complex variable, and define on the basis of this the classes\(L_\beta ^{\psi \mathfrak{N}} (G)\) of (ψ, Β)-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(ζ) 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.