On the class of entire functions defined by Dirichlet series of several complex variables
type: Item
instance of: scholarly article
title: On the class of entire functions defined by Dirichlet series of several complex variables
zbMATH Open document ID: 0760.32005
DOI: 10.1016/0022-247X(91)90195-6
author: Suzanne Meshreky-Daoud
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \(f(s_ 1,s_ 2)=\sum^ \infty_{m,n=1}a_{m,n}\exp(\lambda_ ms_ 1+\mu_ ns_ 2)\) be an entire function defined by Dirichlet series of two complex variables \(s_ 1\) and \(s_ 2\), where the coefficients \(a_{m,n}\) are complex numbers and \[ 0<\lambda_ 1<\lambda_ 2<\dots<\lambda_ m\to\infty\quad\text{for }m\to\infty, \] \[ 0<\mu_ 1<\mu_ 2<\dots<\mu_ n\to\infty\quad\text{for } n\to\infty. \] Let \(X\) be a class of entire functions which satisfies the condition \[ \sum^ \infty_{m,n=1}| a_{m,n}|\exp(m\lambda_ m+n\mu_ n)<\infty. \] For each \(f\in X\), \[ \| f\|=\sum^ \infty_{m,n=1}| a_{m,n}|\exp(m\lambda_ m+n\mu_ n) \] and \(\| f\|\) defines a norm on \(X\). A complex-valued function \(\Phi(f)\) defined for \(f\in X\) is called functional and is said to be linear if \[ \Phi(af+bg)=a\Phi(f)+b\Phi(g). \] Some of the results proved here are: 1. \(X\) is a Banach-algebra. 2. Every continuous linear functional \(\Phi\) on \(X\) is of the form \[ \Phi(f)=\sum^ \infty_{m,n=1}a_{m,n}d_{m,n}\exp(m\lambda_ m+n\mu_ n) \] with \[ f(s_ 1,s_ 2)=\sum^ \infty_{m,n=1}a_{m,n}\exp(\lambda_ ms_ 1+\mu_ ns_ 2), \] where \(\{d_{m,n}\}\) is a bounded sequence.
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 61597
zbMATH Keywords: entire function
zbMATH Keywords: Dirichlet series
reviewed by: Arun Kumar Agarwal
MaRDI profile type: Publication
cites work: Q4722329
cites work: Some class of integral functions represented by Dirichlet series of several complex variables having finite order
cites work: Q3785133
cites work: Q5614773
description: scientific article; zbMATH DE number 61597