Geodesic
On the representation of analytic functions in an open region by Dirichlet series
Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 503-530 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the author's paper (On the representation of analytic functions by Dirichlet series, Mat. Sb. (N.S.), 80(122) (1969), 117–156) a theorem was proved stating that every function $f(z)$, analytic in a finite convex region $D$ and continuous in $\overline D$, can be represented in $D$ by a Dirichlet series. Here we have obtained a definitive result: any function $F(z)$, analytic in $D$, is representable in $D$ by a Dirichlet series. The proof is based on the following assertion. Let $F(z)$ be a function analytic in a finite convex region $D$. There exist a function $f(z)$, analytic in $D$ and continuous in $\overline D$, and an operator $M(y)=\sum_0^\infty c_ny^{(n)}(z)$ with characteristic function $L(\lambda)=\sum_0^\infty c_n\lambda^n$ from the class $[1,0]$, such that $M(f)=F(z)$. Bibliography: 4 titles. 
@article{SM_1970_10_4_a2,
     author = {A. F. Leont'ev},
     title = {On~the representation of analytic functions in an open region by {Dirichlet} series},
     journal = {Sbornik. Mathematics},
     pages = {503--530},
     year = {1970},
     volume = {10},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_4_a2/}
}
TY  - JOUR
AU  - A. F. Leont'ev
TI  - On the representation of analytic functions in an open region by Dirichlet series
JO  - Sbornik. Mathematics
PY  - 1970
SP  - 503
EP  - 530
VL  - 10
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1970_10_4_a2/
LA  - en
ID  - SM_1970_10_4_a2
ER  - 
%0 Journal Article
%A A. F. Leont'ev
%T On the representation of analytic functions in an open region by Dirichlet series
%J Sbornik. Mathematics
%D 1970
%P 503-530
%V 10
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1970_10_4_a2/
%G en
%F SM_1970_10_4_a2
A. F. Leont'ev. On the representation of analytic functions in an open region by Dirichlet series. Sbornik. Mathematics, Tome 10 (1970) no. 4, pp. 503-530. http://geodesic.mathdoc.fr/item/SM_1970_10_4_a2/

[1] A. F. Leontev, “K voprosu o predstavlenii analiticheskikh funktsii ryadami Dirikhle”, Matem. sb., 80(122) (1969), 117–156 | MR

[2] N. Aronszajn, “Sur les decompositions des fonctions analytiques uniformes et sur leurs applications”, Acta Math., 65:1–2 (1935), 1–156 | DOI | MR | Zbl

[3] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, Moskva, 1956

[4] Yu. F. Korobeinik, “Suschestvovanie analiticheskogo resheniya differentsialnogo uravneniya beskonechnogo poryadka i kharakter ego oblasti suschestvovaniya”, Matem. sb., 80(122) (1969), 52–76 | MR | Zbl