Approach properties of funnctions with polinomas in regions in complex plant

In this thesis; the direct and inverse theorems of approximation theory for Dirichlet series in Smirnov spaces over convex polygons have been investigated. The degree of convergence and the regularity of the functions with moduli of arbitrary order have been estimated. Moreover, the influence of differentiably conditions on the rate of approximation and vice versa have been investigated and gives an example on the improvement obtained resuls. Also, in the thesis; the inverse theorem of the approximation theory, about approximation of the functions by polynomials in certain subclasses of Smirnov-Orlicz classes in the domains with Dini-smooth boundaries of the complex plane is proved. Keywords: Dirichlet series, Directand Inverse theorems, Fourier and Leont'ev coefficients, Smirnov spaces, Smirnov-Orlıcz spaces.