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 approximation and vice versa. This work extends results by Yu. I. Mel’nik and gives an example on the improvement by our results. Mathematics Subject Classification (2000): 30 B 50, 41 A 25 Running title: Approximation in Smirnov spaces 1 Dirichlet series Consider the open convex polygon D in the complex plane, with vertices a1, . . . , aN , N > 2. Let D be its closure and ∂D = D \ D the boundary of D. We assume that the origin belongs to D. Let E(D), 1 < p <∞, denote the corresponding Smirnov space; i.e., the Banach space of all functions f(z) which are analytic in D, and satisfy ‖f‖p := sup n∈N ∫ γn |f(z)| |dz| <∞. Here (γn)n∈N is some sequence of closed rectifiable Jordan contours γn ⊂ D which converges to ∂D for n→∞.