Convergent to specific type functions of generalized Faber-Laurent series

SUMMARY Convergent to Specific Type Functions of Generalized Faber and Generalized Faber- Laurent Series The thesis consists of two chapters. In the first chapter, some definitions and theorems related to basis theorems of approximation theory, Bergman spaces, quasiconformal mappings, quasiconformal curves, Faber polynomials and Faber-Laurent series are given. In the second chapter, if G is a continium on the complex plane containing more than one point, and the complement of which with respect to the extended plane being simply connected and containing the point qo, q>:CG-»CD(0,l)is a conformal mapping having conditions 9(00) = oo and lim cp(z)/z > 0, and Gr (R>1) is the finite domain bounded by z- X» TR = {z g C : |cp(z)| = RJ contour line then the approximation problems to the functions of A2(GR) by generalized Faber series of Faber polynomials of G on Gr, l0, for z0eB2, (p2:^2 ->CD(0,1) is a conformal mapping having Z- »co conditions q>2(z0) = qo, lim(z-z0)(p2(z) > 0, TkR := {z : kpk(z)| = Rj, k=l,2; R>1, and Gr is the finite domain bounded by r^R contour lines then generalized Faber-Laurent series for G is defined, this series is uniformly convergent on compact subsets of Gr and the function which is uniformly convergent, belongs to Ho(Gr), the space of functions having zero integral on any closed curves settled in Gr. Finally using Dveirin's integral representations for the functions in Ho(Gr), generalized Faber-Laurent series are defined and it is proved that the generalized Faber-Laurent series of a function /e Ho(Gr) is uniformly convergent to / on compact subsets of Gr. Key Words: Faber Polynomials, Contour Line, Generalized Faber Series, Generalized Faber-Laurent Series.