On Montel's theorem in several variables

Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R,C) to give a new proof of classical Montel's theorem, about continuous solutions of Fréchet's functional equation $\Delta _n^mf = 0$, for real functions (and complex fu...

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Bibliographic Details
Published in:Carpathian Journal of Mathematics 2015-01, Vol.31 (1), p.1-10
Main Authors: Almira, J. M., Abu-Helaiel, Kh. F.
Format: Article
Language:English
Subjects:
Complex numbers
Degrees of polynomials
Geometric translations
Linear transformations
Mathematical functions
Mathematical theorems
Mathematical vectors
Polynomials
Real variables
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Summary:Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R,C) to give a new proof of classical Montel's theorem, about continuous solutions of Fréchet's functional equation $\Delta _n^mf = 0$, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel's type theorem for the case of complex valued functions defined over the discrete group Zd. Furthermore, we also state and demonstrate an improved version of Montel's Theorem for complex functions of several real variables and complex functions of several complex variables.
ISSN:1584-2851
1843-4401
DOI:10.37193/CJM.2015.01.01