Approximation in Smirnov spaces: Direct and inverse theorems

We give direct and inverse approximation theorems for Dirichlet series in Smirnov spaces over convex polygons. We estimate the degree of convergence and the regularity of the functions with moduli of arbitrary order k. Moreover, we consider the influence of differentiability conditions on the rate of pproximation and vice versa. This work extends results by Yu. I. Mel'nik and gives an example on the improvement by our results.