Geodesic
Extension of entire functions of completely regular growth and right inverse to the operator of representation of analytic functions by quasipolynomial series
Sbornik. Mathematics, Tome 191 (2000) no. 7, pp. 1049-1073 
S. N. Melikhov. Extension of entire functions of completely regular growth and right inverse to the operator of representation of analytic functions by quasipolynomial series. Sbornik. Mathematics, Tome 191 (2000) no. 7, pp. 1049-1073. http://geodesic.mathdoc.fr/item/SM_2000_191_7_a5/
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     author = {S. N. Melikhov},
     title = {Extension of entire functions of completely regular growth and right inverse to the~operator of representation of analytic functions by quasipolynomial series},
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Let $L$ be an entire function of one complex variable that has exponential type, completely regular growth, and whose conjugate diagram is equal to the sum of the closure of a bounded convex domain $G$ and a convex compact subset $K$ of $\mathbb C$. Criteria ensuring that the operator $R$ of the representation of analytic functions in $G$ by quasipolynomial series with zeros of the function $L$ as exponents has a continuous linear right inverse are established. These criteria are stated in terms of conformal maps of the unit disc onto the domain $G$ and of the exterior of the closed unit disc onto the exterior of $K$, and of extensions of the original function $L$ to an entire function $Q$ of two complex variables whose absolute value satisfies certain (upper) estimates. An analogue of the Leont'ev interpolation function defined by this extension $Q$ is used to obtain formulae for the continuous linear right inverse to the representation operator $R$.

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

[2] Krasichkov-Ternovskii I. F., “Odna geometricheskaya lemma, poleznaya v teorii tselykh funktsii, i teoremy tipa Levinsona”, Matem. zametki, 24:4 (1978), 531–546 | MR

[3] Korobeinik Yu. F., “Predstavlyayuschie sistemy podprostranstv”, Matem. zametki, 38:5 (1985), 741–755 | MR | Zbl

[4] Napalkov V. V., Komarov A. V., O razlozhenii analiticheskikh v polioblasti funktsii v ryady po elementarnym resheniyam sistem uravnenii svertki, Preprint, Ufa, 1989 | Zbl

[5] Melichow S. N., “Über absolut repräsentierende Systeme aus Quasipolynomen in Räumen analytischer Funktionen”, Math. Nachr., 158 (1992), 299–323 | MR | Zbl

[6] Korobeinik Yu. F., Melikhov S. N., “Lineinyi nepreryvnyi pravyi obratnyi dlya operatora predstavleniya i konformnye otobrazheniya”, Dokl. RAN, 323:5 (1992), 826–829 | MR | Zbl

[7] Korobeinik Yu. F., Melikhov S. N., “Lineinyi nepreryvnyi pravyi obratnyi dlya operatora predstavleniya i prilozheniya k operatoram svertki”, Sib. matem. zhurn., 34:1 (1993), 70–84 | MR | Zbl

[8] Meise R., “Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals”, J. Reine Angew. Math., 363 (1985), 59–95 | MR | Zbl

[9] Momm S., “Convolution equations on the analytic functions on convex domains in the plane”, Bull. Sci. Math. (2), 345 (1994), 729–752 | MR | Zbl

[10] Melikhov S. N., Momm S., “Solution operators for convolution equations on the germs of analytic functions on compact convex sets of $\mathbb C^N$”, Studia Math., 117:1 (1995), 79–99 | MR | Zbl

[11] Melikhov S. N., Momm Z., “O lineinom nepreryvnom pravom obratnom dlya operatora svertki na prostranstvakh rostkov analiticheskikh funktsii na vypuklykh kompaktakh v $\mathbb C$”, Izv. vuzov. Ser. matem., 1997, no. 5, 38–48 | MR | Zbl

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

[13] Momm S., “Convex univalent functions and continuous linear right inverses”, J. Funct. Anal., 103 (1992), 85–103 | DOI | MR | Zbl

[14] Momm S., “Extremal plurisubharmonic functions for convex bodies in $\mathbb C^N$”, Complex analysis, harmonic analysis and applications, Proceedings of a conference in honour of the retirement of Roger Gay (June 7–9, 1995, Bordeaux, France), eds. R. Deville et al., Longman, Harlow, 1996, 87–103 | MR | Zbl

[15] Korobeinik Yu. F., “Netrivialnye razlozheniya nulya po absolyutno predstavlyayuschim sistemam. Prilozheniya k operatoram svertki”, Matem. sb., 182:5 (1991), 661–680 | MR

[16] Abanin A. V., “Netrivialnye razlozheniya nulya i absolyutno predstavlyayuschie sistemy”, Matem. zametki, 57:4 (1995), 483–497 | MR | Zbl

[17] Edvards R., Funktsionalnyi analiz. Teoriya i prilozheniya, Mir, M., 1969

[18] Khermander L., Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[19] Melikhov S. N., “O razlozhenii analiticheskikh funktsii v ryady eksponent”, Izv. AN SSSR. Ser. matem., 52:5 (1988), 991–1004 | Zbl

[20] Lelon P., Gruman L., Tselye funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1989 | MR | Zbl

[21] Korobeinik Yu. F., “O resheniyakh nekotorykh funktsionalnykh uravnenii v klassakh funktsii, analiticheskikh v vypuklykh oblastyakh”, Matem. sb., 75 (117):2 (1968), 225–234 | MR | Zbl

[22] Korobeinik Yu. F., “Operatory svertki v kompleksnoi oblasti”, Matem. sb., 127:2 (1985), 173–197 | MR | Zbl

[23] Korobeinik Yu. F., “O multiplikatorakh vesovykh funktsionalnykh prostranstv”, Analysis Math., 15:105–114 (1989) | MR | Zbl

[24] Leontev A. F., “Ob ekvivalentnykh usloviyakh predstavleniya analiticheskikh funktsii ryadami eksponent”, Matem. zametki, 20:1 (1976), 91–104 | MR | Zbl

[25] Markushevich A. I., Teoriya analiticheskikh funktsii, T. 1, Nauka, M., 1967

[26] Jarchow H., Locally convex spaces, Teubner, Stuttgart, 1981 | MR | Zbl

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

[28] Shabat B. V., Vvedenie v kompleksnyi analiz, Nauka, M., 1969 | MR | Zbl

[29] Momm S., “On the dependence of analytic solutions of partial differential equations from the right-hand side”, Trans. Amer. Math. Soc., 345 (1994), 729–752 | DOI | MR | Zbl

[30] Momm S., “A critical growth rate of the pluricomplex Green function”, Duke Math. J., 72 (1993), 487–502 | DOI | MR | Zbl

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