Direct approximation theorems for Dirichlet series in the norm of uniform convergence

We consider functions f∈AC(D¯) on a convex polygon D⊂C and their regularity in terms of Tamrazov's moduli of smoothness. Using the relation between Fourier and Leont'ev coefficients given in (CMFT 1(1) (2001) 193) we prove direct approximation theorems of Jackson type for the Dirichlet expansionf(z)∼∑λ∈Λκf(λ)eλzL′(λ),where L(z)=∑k=1Ndkeakz is a quasipolynomial with respect to the vertices a1,…,aN of D and Λ its set of zeros. We show by an example that our results improve Mel'nik's estimates in (Ukrainian Math. J. 40(4) (1988) 382) on the rate of convergence.