Maximal approximation's with polynomials in the complex plane

This master's thesis consists of an introduction, preliminary information section, and three main chapters. In the introduction, research conducted on the thesis topic and the results obtained in this area are presented. The preliminary information section contains the symbols, some definitions, and auxiliary results used in the thesis. In the third chapter, starting from an integral representation that exists in weighted Bergman spaces defined on quasiconformal regious bounded by quasiconformal curves the Faber series expansions of functions are obtained, and the problem of maximal convergence in continua is investigated using partial sums of the series. In the fourth chapter, the problem of maximal convergence in the Smirnov-Orcliz classes of analytic functions is studied, the smoothness moduli in these classes are defined and the maximal convergence error in terms of the smoothness moduli and best approximation number are estimated. In the fifth chapter, variable exponent Smirnov Classes in complex plane regions are defined, the possibility of Faber series expansions in these classes is shown, the maximal convergence error of the partial sums of Faber series expansions of given function in terms of the best approximation number is estimated and later using the direct theorems of approximation theory this error by the smoothness modulus are evaluated.