On the class of entire functions defined by Dirichlet series of several complex variables
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.
scientific article; zbMATH DE number 61597