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On the construction of entire and meromorphic functions of several variables having specified growth, and some applications

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Název: On the construction of entire and meromorphic functions of several variables having specified growth, and some applications
Autoři: Sekerin, A. B.
Informace o vydavateli: American Mathematical Society (AMS), Providence, RI
Témata: representation of analytic functions in several variables by exponential series, meromorphic function, Special classes of entire functions of one complex variable and growth estimates, Other generalizations of function theory of one complex variable, Meromorphic functions of one complex variable (general theory), approximation, Harmonic, subharmonic, superharmonic functions in higher dimensions, subharmonic functions
Popis: The author generalizes a result concerning the approximation of differences of certain subharmonic functions in \(\mathbb{C} = \mathbb{R}^2\) by the logarithm of the modulus of a meromorphic function to the higher dimensional case. The main result is: Let \(u\) be the difference of subharmonic functions in \(\mathbb{C}^n\) such that \[ |u(z) |\leq c (\varepsilon) |z |^{\rho + \varepsilon} \] \(0 < \rho < 2\), for any \(\varepsilon > 0\) outside a countable union of balls with finite sum of radii, then there are entire functions \(L_1\), \(L_2\) such that outside a \(C_0\)-set \[ \bigl |u(z) - \ln |L_1 (z)/L_2 (z) |\bigr |\leq C \bigl( \ln |z |\bigr)^3 |z |^{\rho (1 - 1/2n}). \] This result is applied to the representation of analytic functions in several variables by exponential series.
Druh dokumentu: Article
Popis souboru: application/xml
Přístupová URL adresa: https://zbmath.org/738900
Přístupové číslo: edsair.c2b0b933574d..2a09b0ef0e9dce7e2c3c2c88e39b2bbe
Databáze: OpenAIRE
Popis
Abstrakt:The author generalizes a result concerning the approximation of differences of certain subharmonic functions in \(\mathbb{C} = \mathbb{R}^2\) by the logarithm of the modulus of a meromorphic function to the higher dimensional case. The main result is: Let \(u\) be the difference of subharmonic functions in \(\mathbb{C}^n\) such that \[ |u(z) |\leq c (\varepsilon) |z |^{\rho + \varepsilon} \] \(0 < \rho < 2\), for any \(\varepsilon > 0\) outside a countable union of balls with finite sum of radii, then there are entire functions \(L_1\), \(L_2\) such that outside a \(C_0\)-set \[ \bigl |u(z) - \ln |L_1 (z)/L_2 (z) |\bigr |\leq C \bigl( \ln |z |\bigr)^3 |z |^{\rho (1 - 1/2n}). \] This result is applied to the representation of analytic functions in several variables by exponential series.