Bergman's polynomials and their some properties

"Bergman's polynomials and their some properties" This study entitled as "Bergman's polynomials and their some properties" consists of three chapters. The first chapter as a preliminary from of the next chapters includes some definition and theorems (with or without proof) concerning the analytic functions, curves and the families of curves in the complex plane. The absolutely continuous, differentiable, LP- differentiable and integrals of the complex valued functions; Cauchy and Cauchy Pompeiu's formulas are given. In the second chapter, A2(G) function space is introduced and it is proved that this space is a Hubert space. {gk(z)}, k = 0,1,2,... which is orthonormal system in A2(G) is constructed from the {fk(z)}, k = 0,1,2,... which is linearly independent system in A2(G) by the Gram-Scmidt orthogonalization process. Then a special case {K"(z)} Bergman's polynomials that are orthonormal in G are constructed. Some elementary properties of these polynomials are presented (their zeros and external). As in A2(G), A2(y,G) space with y(z) weight function is introduced, and orthonormal polynomials with respect to y(z) weight function are given. The third chapter starts with an introduction. Then the problem as the main objective of the study is established. Having given the theorems of Riemann mapping, the quasiconformal mappings, the quasiconformal curves, the module of families of curves, some local properties of the function of Riemann mapping, two theorems bear on the approximation of special function are given in A2(G) space. Finally, by which speed Bergman's polynomials with weight function in domain G are approaching to 0 for interior point of the domain is found regarding to the properties of G and weight function. Key Words: An analytic function, a conformal mapping, a quasiconformal mapping, a quasiconformal curve, a quasiconformal reflection, orthonormal system, a weight function, orthonormal (Bergman) polynomials with a weight function. v