Geodesic
Nontrivial expansions of zero in absolutely representing systems. Application to convolution operators
Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 49-66 
Yu. F. Korobeinik. Nontrivial expansions of zero in absolutely representing systems. Application to convolution operators. Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 49-66. http://geodesic.mathdoc.fr/item/SM_1992_73_1_a3/
@article{SM_1992_73_1_a3,
     author = {Yu. F. Korobeinik},
     title = {Nontrivial expansions of zero in absolutely representing systems. {Application} to convolution operators},
     journal = {Sbornik. Mathematics},
     pages = {49--66},
     year = {1992},
     volume = {73},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_73_1_a3/}
}
TY  - JOUR
AU  - Yu. F. Korobeinik
TI  - Nontrivial expansions of zero in absolutely representing systems. Application to convolution operators
JO  - Sbornik. Mathematics
PY  - 1992
SP  - 49
EP  - 66
VL  - 73
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1992_73_1_a3/
LA  - en
ID  - SM_1992_73_1_a3
ER  - 
%0 Journal Article
%A Yu. F. Korobeinik
%T Nontrivial expansions of zero in absolutely representing systems. Application to convolution operators
%J Sbornik. Mathematics
%D 1992
%P 49-66
%V 73
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1992_73_1_a3/
%G en
%F SM_1992_73_1_a3

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

By using a general representation of nontrivial expansions of zero in absolutely representing systems of the form $\{E_\rho(\lambda_kz)\}_{k=1}^\infty$, where $\rho>0$, $E_\rho(z)=\sum\limits_{n=0}^\infty\dfrac{z^n}{\Gamma(1+\frac n\rho)}$ is the Mittag-Leffler function, and $(\lambda_k)_{k=1}^\infty$ are complex numbers, the author obtains a number of results in the theory of $\rho$-convolution operators in spaces of functions that are analytic in $\rho$-convex domains (a description of the general solution of a homogeneous $\rho$-convolution equation and of systems of such equations, a topological description of the kernel of a $\rho$-convolution operator, the construction of principal solutions, and a criterion for factorization).

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

[2] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966 | Zbl

[3] Epifanov O. V., Lenev A. A., “O razreshimosti odnogo integralnogo uravneniya v prostranstvakh analiticheskikh funktsii”, Matem. analiz i ego pril., 6, Izd-vo Rost. un-ta, Rostov n/D, 1974, 258–261

[4] Tkachenko V. A., “Uravneniya tipa svertki v prostranstvakh analiticheskikh funktsionalov”, Izv. AN SSSR. Ser. matem., 41:2 (1977), 378–391 | MR | Zbl

[5] Korobeinik Yu. F., “Ob odnoi interpolyatsionnoi zadache dlya tselykh funktsii”, Izv. VUZov. Matem., 1985, no. 2, 37–45 | MR | Zbl

[6] Vogt D., “Eine Charakterisierungen der Potenzreihenräume von endlichem Typ und ihre Folgerungen”, Manuscripta Math., 37 (1982), 269–301 | DOI | MR | Zbl

[7] Meise R., Taylor B. A., “Sequence space representation for (FN)-algebras of entire functions modulo closed ideals”, Stud. Mathem., LXXXV (1987), 203–227 | MR | Zbl

[8] Pich A., Yadernye lokalno vypuklye prostranstva, Mir, M., 1967 | MR

[9] Leontev A. F., Ryady eksponent, Nauka, M., 1976 | MR

[10] Korobeinik Yu. F., “Uravneniya svertki v kompleksnoi oblasti”, Matem. sb., 127(169) (1985), 173–196 | MR

[11] Korobeinik Yu. F., “Predstavlyayuschie sistemy”, UMN, 36:1 (1981), 73–126 | MR | Zbl

[12] Korobeinik Yu. F., “Interpolyatsionnaya zadacha, netrivialnye razlozheniya nulya i predstavlyayuschie sistemy”, Izv. AN SSSR. Ser. matem., 44:5 (1980), 1066–1114 | MR | Zbl

[13] Azarin V. S., “Ob odnom kharakteristicheskom svoistve funktsii vpolne regulyarnogo rosta vnutri ugla”, Teoriya funktsii, funktsion. analiz i ikh pril., 2 (1966), 55–66 | MR | Zbl

[14] Robertson A., Robertson V., Topologicheskie vektornye prostranstva, Mir, M., 1967 | MR | Zbl

[15] Napalkov V. V., “O bazise v prostranstve reshenii uravneniya svertki”, Matem. zametki, 43:1 (1988), 44–55 | MR | Zbl | Zbl

[16] Bratischev A. V., Posledovatelnosti s konechnoi maksimalnoi uglovoi plotnostyu i nekotorye ikh prilozheniya, Dep. v VINITI 14.12.87. No 8703-87, 90 pp.

[17] Valiron G., “Sur les solutions des équations différentielles linéaires d'ordre infini et a coefficients constants”, Ann. Sci. Ecole Norm. Sup., 46:1 (1929), 25–53 | MR | Zbl

[18] Polya G., “Eine Verallegemeinerung des Fabryschen Lückensatzes”, Nachr. Gesell. Wiss. Göttingen, 1927, 187–195 | Zbl

[19] Krasichkov-Ternovskii I. F., “Invariantnye podprostranstva analiticheskikh funktsii. II: Spektralnyi analiz na vypuklykh oblastyakh”, Matem. sb., 88 (130) (1972), 3–29 | MR | Zbl

[20] Znamenskii S. V., Ob oblastyakh suschestvovaniya analiticheskikh reshenii differentsialnogo uravneniya beskonechnogo poryadka s postoyannymi koeffitsientami, Preprint IFSO-6M, In-t fiziki im. L. V. Kirenskogo SO AN SSSR, Krasnoyarsk, 1976

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

[22] Timofeev A. Yu., “O predstavlenii resheniya uravneniya beskonechnogo poryadka v vide summy dvukh reshenii”, Matem. zametki, 31:2 (1982), 245–256 | MR | Zbl

[23] Timofeev A. Yu., “O predstavlenii resheniya operatornogo uravneniya v vide summy reshenii”, Issledovaniya po teorii operatorov, BF AN SSSR, Ufa, 1988, 146–152

[24] Napalkov V. V., Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR | Zbl

[25] Leontev A. F., Posledovatelnosti polinomov iz eksponent, Nauka, M., 1976 | MR

[26] Korobeinik Yu. F., “Operatornye uravneniya beskonechnogo poryadka s postoyannymi koeffitsientami”, SMZh, XVII:3 (1976), 571–585 | MR | Zbl