Classes of $(ψ, β)$-differential functions of complex variable and approximation by linear averages of their Faber series - Stepanec, A.I.; Romanyuk, V.S. (Ukrains’kyi Matematychnyi Zhurnal, 1992)
We introduce the notion of $(ψ, β)$-derivative of a function of one complex variable, and define on the basis of this the classes $L_β ^{ψ}{\mathfrak N} (G)$ of $(ψ, β)$-differentiable analytic functions in a bounded domain $G$. The classes $L_β ^{ψ}{\mathfrak N} (G)$ 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 $L_β ^{ψ}{\mathfrak N}$. We consider approximation of functions $f\in L_β ^{ψ}{\mathfrak N} (G)$ by algebraic polynomials constructed from their series expansions in Faber polynomials.