The maximal convergence of fourier series with bergman orthogonal polynomials

The aim of the thesis, is to search properties of the convergence and maximal convergence of the Fourier series of a function in the Bergman spaces, which is constructed with the Bergman orthogonal polynomials of the domain. The thesis consists of six chapters. In the first chapter, the knowledge about the problems and the developments of the approximation theory are given. In the second chapter, the fundamental definitions and theorems and some examples are included. In the third chapter, the definition of Bergman space and some lemmas and theorems according to the inner product of this space are given. In the fourth chapter, the Bergman orthogonal polynomials are introduced and the expansions to the ON systems in a Hilbert space and the Bergman space are investigated. In the fifth chapter, firstly the classical theorems expressing the possibility of approximation with algebraic polynomials in the complex plane are examined, and then the convergence of Fourier series with respect to the orthonormal polynomials of the domain is investigated, some theorems in which the rate of convergence is estimated in the case of some domains are given. In the sixth chapter, firstly Bernstein & Walsh lemma on the rate of growth of polynomials on a larger domain via to the maximal value of their in the closed domain are given. Finally, the maximal convergence of Fourier series of an analytic function f in the canonical domain, constructed with orthonormal polynomials of the domain is examined in the Walsh's theorem in case of closure of the region and in the Suetin's theorem in case of closure of the canonical region.