Geodesic
Summability of the Dirichlet series with real exponents for an arbitrary analytic function
Izvestiya. Mathematics , Tome 1 (1967) no. 1, pp. 81-94.

Voir la notice de l'article provenant de la source Math-Net.Ru

Given a function $f(z)$ that is analytic on a closed vertical interval of length $2\pi\sigma$, we use a definite rule to associate with it a formal Dirichlet series with exponents $\pm\lambda_n(n=1,2,\dots)$, where $\lambda_n>0$ and $\lim\limits_{n\to\infty}\frac{n}{\lambda_n}=\sigma$. In general this series diverges everywhere. We give a method for summing it to the function $f(z)$. 
@article{IM2_1967_1_1_a5,
     author = {A. F. Leont'ev},
     title = {Summability of the {Dirichlet} series with real exponents for an arbitrary analytic function},
     journal = {Izvestiya. Mathematics },
     pages = {81--94},
     publisher = {mathdoc},
     volume = {1},
     number = {1},
     year = {1967},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1967_1_1_a5/}
}
TY  - JOUR
AU  - A. F. Leont'ev
TI  - Summability of the Dirichlet series with real exponents for an arbitrary analytic function
JO  - Izvestiya. Mathematics 
PY  - 1967
SP  - 81
EP  - 94
VL  - 1
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1967_1_1_a5/
LA  - en
ID  - IM2_1967_1_1_a5
ER  - 
%0 Journal Article
%A A. F. Leont'ev
%T Summability of the Dirichlet series with real exponents for an arbitrary analytic function
%J Izvestiya. Mathematics 
%D 1967
%P 81-94
%V 1
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1967_1_1_a5/
%G en
%F IM2_1967_1_1_a5
A. F. Leont'ev. Summability of the Dirichlet series with real exponents for an arbitrary analytic function. Izvestiya. Mathematics , Tome 1 (1967) no. 1, pp. 81-94. http://geodesic.mathdoc.fr/item/IM2_1967_1_1_a5/

[1] Leontev A. F., “O polnote sistemy $\{e^{\lambda_{n^z}}\}$ v zamknutoi polose”, Dokl. AN SSSR, 152:2 (1963), 266–268 | MR

[2] Schwartz L., Etude des sommes d'exponentielles, deuxième èd., Actualites Scientifiques et Industrieiles, 1959

[3] Leontev A. F., “O svoistvakh posledovatelnostei polinomov Dirikhle, skhodyaschikhsya na intervale mnimoi osi”, Izv. AN SSSR. Ser. matem., 29 (1965), 269–328 | MR

[4] Bernstein V., Legons sur les progrès récents de la théorie des séries de Dirichlet, Paris, 1933

[5] Leontev A. F., “O skhodimosti posledovatelnosti polinomov Dirikhle”, Dokl. AN SSSR, 108:2 (1956), 23–26 | MR

[6] Leontev A. F., “Rasprostranenie svoistv tselykh funktsii poryadka menshe poloviny na nekotorye drugie funktsii”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, LXIV, 1961, 126–146 | MR

[7] Baillette A., “Approximation de fonctions par des sommeis d'exponentielles”, Gompt. rend. Acad. Sci., 249:23 (1959), 2470–2471 | MR | Zbl

[8] Baillette A., “Fonctions appraohables par des sommeis d'exponentielles”, J. analyse math., 10:2 (1962–1963), 91–114 | DOI | MR

[9] Krasichkov I. F., “Ob otsenkakh snizu dlya tselykh funktsii konechnogo poryadka i o skhodimosti polinomov Dirikhle”, Dokl. AN SSSR, 162:5 (1965), 995–996 | Zbl

[10] Smirnov V. I., Kurs vysshei matematiki, t. III, ch. 2, GITTL, M., 1950