Geodesic
On the representation by Dirichlet series of analytic functions in a~closed convex polygonal region
Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 133-144.

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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. 
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A. F. Leont'ev. On the representation by Dirichlet series of analytic functions in a~closed convex polygonal region. Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 133-144. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a7/

[1] Leontev A. F., “O predstavlenii analiticheskikh funktsii v zamknutoi vypukloi oblasti ryadami Dirikhle”, Izv. AN SSSR. Ser. matem., 37 (1973), 577–592 | MR

[2] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[3] Gelfond A. O., “O polnote sistem analiticheskikh funktsii”, Matem. sb., 4(46) (1938), 149–156 | Zbl