Zero sequences of holomorphic functions, representation

B. N. Khabibullin

Sbornik. Mathematics, Tome 198 (2007) no. 2, pp. 261-298

Résumé

Let $\Lambda=\{\lambda_k\}$ be a point sequence in a subdomain $\Omega$ of the complex plane $\mathbb C$. In terms of harmonic measures, Green's functions, balayage, Jensen measures, and so on, general conditions are described ensuring that $\Lambda$ is the zero sequence of a holomorphic function in a prescribed weighted space of holomorphic functions in $\Omega$. The question of the representation of a meromorphic function in $\Omega$ as the ratio of holomorphic functions without common zeros from a prescribed weighted space is considered in similar terms. Some applications are presented. Bibliography: 46 titles.

Bibliographie

[1] B. N. Khabibullin, Zero (sub)sets for spaces of holomorphic functions and (sub)harmonic minorants, arXiv: math.CV/0412359

[2] M. M. Dzhrbashyan, “O kanonicheskom predstavlenii meromorfnykh v edinichnom kruge”, Dokl. AN ArmSSR, 3:1 (1945), 3–9

[3] M. M. Dzhrbashyan, “Teoriya faktorizatsii funktsii, meromorfnykh v kruge”, Matem. sb., 79(121):4(8) (1969), 517–615 | MR | Zbl

[4] M. M. Dzhrbashyan, “Teoriya faktorizatsii i granichnykh svoistv funktsii, meromorfnykh v kruge”, UMN, 28:4(172) (1973), 3–14 | MR | Zbl

[5] P. Colwell, Blaschke product. Bounded analytic functions, Univ. Michigan Press, Ann Arbor, MI, 1985 | MR

[6] C. Horowitz, “Zeros of functions in the Bergman spaces”, Duke Math., 41 (1974), 693–710 | DOI | MR | Zbl

[7] E. Beller, “Factorization for non-Nevanlinna classes of analytic functions”, Israel J. Math., 27:3–4 (1977), 320–330 | DOI | MR | Zbl

[8] E. Beller, C. Horowitz, “Zero sets and random zero sets in certain function spaces”, J. Anal. Math., 64 (1994), 203–217 | DOI | MR | Zbl

[9] B. Korenblum, “An extension of the Nevanlinna theory”, Acta Math., 135 (1975), 187–219 | DOI | MR | Zbl

[10] F. A. Shamoyan, “O nulyakh analiticheskikh v kruge funktsii, rastuschikh vblizi granitsy”, Izv. AN ArmSSR. Ser. matem., 18:1 (1983), 15–27 | MR | Zbl

[11] M. M. Dzhrbashyan, “Faktorizatsiya funktsii, meromorfnykh v konechnoi ploskosti”, Izv. AN ArmSSR. Ser. matem., 5:6 (1970), 453–485 | MR | Zbl

[12] A. A. Goldberg, B. Ya. Levin, I. V. Ostrovskii, “Tselye i meromorfnye funktsii”, Itogi nauki i tekhniki. Sovrem. problemy matem. Fundam. napr., 85, VINITI, M., 1991, 5–185 ; A. A. Gol'dberg, B. Ya. Levin, I. V. Ostrovkii, “Entire and meromorphic functions”, Complex analysis I. Encyclopaedia Math. Sci., 85, 1995, 1–193 | MR | Zbl | Zbl

[13] Ya. V. Vasilkiv, Yu. S. Protsik, “Kanonichnii integral Veiershtrassa dlya subgarmoniinikh funktsii neskinchennogo poryadku”, Visnik Kharkivskogo natsionalnogo universitetu Ser. matem., prikl. matem., mekh., 542:51 (2002), 118–130 | Zbl

[14] A. A. Kondratyuk, Ya. V. Vasyl'kiv, “Growth majorants and quotient representations of meromorphic functions”, Comput. Metods Funct. Theory, 1:2 (2001), 595–606 | MR | Zbl

[15] B. N. Khabibullin, “The representation of a meromorphic function as the quotient of entire functions and Paley problem in $\mathbb C^n$: survey of some results”, Mat. Fiz. Anal. Geom., 9:2 (2002), 146–167 | MR | Zbl

[16] B. N. Khabibullin, “Dual approach to certain questions for the weighted spaces of holomorphic functions”, Entire functions in modern analysis (Tel-Aviv, 1997), 15, 2001, 207–219 | MR | Zbl

[17] B. N. Khabibullin, “Dvoistvennoe predstavlenie superlineinykh funktsionalov i ego primeneniya v teorii funktsii. I”, Izv. RAN. Ser. matem., 65:4 (2001), 205–224 | MR | Zbl

[18] B. N. Khabibullin, “Dvoistvennoe predstavlenie superlineinykh funktsionalov i ego primeneniya v teorii funktsii. II”, Izv. RAN. Ser. matem., 65:5 (2001), 167–190 | MR | Zbl

[19] B. N. Khabibullin, “Tselye funktsii-minoranty: opyt primeneniya otsenok Matsaeva–Ostrovskogo–Sodina”, Matem. fizika, analiz, geometriya, 11:4 (2004), 518–536 | MR | Zbl

[20] B. N. Khabibullin, “Nulevye podmnozhestva, predstavlenie meromorfnykh funktsii i kharakteristiki Nevanlinny v kruge”, Matem. sb., 197:2 (2006), 117–136 | MR

[21] B. N. Khabibullin, Polnota sistem eksponent i mnozhestva neeedinstvennosti, 14, RITs BashGU, Ufa, 2006 | MR

[22] B. N. Khabibullin, “Zero (sub)sets for weighted spaces of holomorphic functions”, Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz, Mezhdunarodnaya konferentsiya, posvyaschennaya stoletiyu S. M. Nikolskogo (Moskva, 2005), 2005, 307

[23] B. N. Khabibullin, “On the representation of meromorphic functions”, Analysis and related topics (Lviv, 2005), Lviv Ivan Franko Natiuonal Univ., Lviv, 2005, 48

[24] U. Kheiman, P. Kennedi, Subgarmonicheskie funktsii, Mir, M., 1980 ; W. K. Hayman, P. B. Kennedy, Subharmonic functions, vol. I, London Math. Soc. Monogr., 9, Academic Press, London, New York, 1976 | MR | Zbl | MR | Zbl

[25] T. J. Ransford, Potential theory in the complex plane, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[26] L. Shvarts, Analiz, t. 1, Mir, M., 1972; L. Schwartz, Analyse mathématique, Hermann, Paris, 1967 | MR | Zbl | Zbl

[27] N. Danford, Dzh. Shvarts, Lineinye operatory. Obschaya teoriya, IL, M., 1962 ; N. Dunford, J. T. Schwartz, Linear operators. I. General theory, Interscience Publ., New York, London, 1958 | MR | MR | Zbl

[28] T. W. Gamelin, Uniform algebras and Jensen measures, Cambridge Univ. Press, Cambridge, 1978 | MR | Zbl

[29] B. N. Khabibullin, “Kriterii (sub-)garmonichnosti i prodolzhenie (sub-)garmonicheskikh funktsii”, Sib. matem. zh., 44:4 (2003), 905–925 | MR | Zbl

[30] S. A. Grigoryan, “Obobschennye analiticheskie funktsii”, UMN, 49:2 (1994), 3–42 | MR | Zbl

[31] M. Klimek, Pluripotential theory, Clarendon Press, Oxford, 1991 | MR | Zbl

[32] A. G. Naftalevich, “Ob interpolirovanii funktsii, meromorfnykh v edinichnom kruge”, Dokl. AN SSSR, 88:2 (1953), 205–208 | MR | Zbl

[33] A. G. Naftalevich, “Ob interpolirovanii funktsii, meromorfnykh v edinichnom kruge”, Litovskii matem. sb., 1:1–2 (1961), 159–180 | MR | Zbl

[34] M. Tsuji, “Canonical product for a meromorphic function in a unit circle”, J. Math. Soc. Japan, 8:1 (1956), 7–21 | MR | Zbl

[35] G. Bomash, “A Blashke-type product and random zero sets for Bergman spaces”, Ark. Math., 30 (1992), 45–60 | DOI | MR | Zbl

[36] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Tokyo, 1959 | MR | Zbl

[37] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York, 1992 | MR | Zbl

[38] B. N. Khabibullin, “O tipe tselykh i meromorfnykh funktsii”, Matem. sb., 183:11 (1992), 35–44 ; B. N. Khabibullin, “On the type of entire and meromorphic functions”, Russian Acad. Sci. Sb. Math., 77:2 (1994), 293–301 | MR | Zbl | DOI

[39] T. J. Ransford, “Jensen measures”, Approximation, complex analysis and potential theory (Montréal, QC, 2000), Kluwer, Dordrecht, 2001, 221–237 | MR | Zbl

[40] P. Koosis, Leçons sur le théorème de Beurling et Malliavin, CRM, Montréal, 1996 | MR | Zbl

[41] Ü. Kuran, “On measures associated to superharmonic functions”, Proc. Amer. Math. Soc., 36:1 (1972), 179–186 | DOI | MR | Zbl

[42] C. Sundberg, “Measures induced by analytic functions and a problem of Walter Rudin”, J. Amer. Math. Soc., 16:1 (2003), 69–90 | DOI | MR | Zbl

[43] N. S. Landkof, Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 ; N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, Heidelberg, 1972 | MR | Zbl | MR | Zbl

[44] B. J. Cole, T. J. Ransford, “Jensen measures and harmonic measures”, J. Reine Angew. Math., 541 (2001), 29–53 | MR | Zbl

[45] B. J. Cole, T. J. Ransford, “Subharmonicity without upper semicontinuity”, J. Funct. Anal., 147 (1997), 420–442 | DOI | MR | Zbl

[46] N. Burbaki, Topologicheskie vektornye prostranstva, IL, M., 1959; N. Bourbaki, Elements of mathematics. Topological vector spaces, Springer-Verlag, Berlin, 1987 | MR | Zbl