On the representation by Dirichlet series of analytic functions in a closed convex polygonal region

Let $\overline D$ be a closed convex polygonal region. It is shown that, for any function $f(z)$ analytic in the open region $D$ and continuous together with its first derivative in $\overline D$, a Dirichlet series can be constructed (its exponents depend only on $D$) that converges to $f(z)$ everywhere in $\overline D$ except, possibly, at its vertices.