Treffer: Bases in the space of entire functions of several complex variables represented by Dirichlet series having finite order point

Let \(\rho_ 1\), \(\rho_ 2\) be arbitrary positive numbers and \(\lambda_ 0=\mu_ 0=0\), \(\{\lambda_ m\}_{m\geq 1}\), \(\{\mu_ n\}_{n\geq 1}\) be two increasing sequences of real numbers such that \[ \lim_{m\to+\infty}\lambda_ m=\lim_{n\to+\infty}\mu_ n=+\infty, \] \[ \limsup_{m+n\to+\infty}(\lambda_ m+\mu_ n)^{- 1}\cdot\log(m+n)0\) \(\sigma_ 1,\sigma_ 2\geq A=A(\varepsilon)\). Here \[ M_ f(\sigma_ 1,\sigma_ 2):=\sup_{-\infty0}\) on \(Y\), where \[ \| f;\rho_ 1+\delta,\rho_ 2+\delta\|\equiv| a_{0,0}|+\sum_{m+n\geq 1}| a_{m,n}|\lambda_ m^{\lambda_ m\cdot(\rho_ 1+\delta)^{-1}}\mu_ n^{\mu_ n\cdot (\rho_ 2+\delta)^{-1}}, \] \(\delta>0\), for an arbitrary function \(f\in Y\) of the form (1). Definition 1: Let \(Y_ 0\) be a closed subspace of \(Y\). A sequence \(\{f_{m,n}\}^{+\infty}_{m,n=0}\subset Y\) is called a basis in \(Y_ 0\) if every element \(f\in Y_ 0\) is uniquely expressed as \(f=\sum^{+\infty}_{m,n=0}a_{m,n}\cdot f_{m,n}\), \(\{a_{m,n}\}^{+\infty}_{m,n=0}\subset\mathbb{C}\), where convergence of the series is with respect to the topology on \(Y\). Definition 2: A basis \(\{f_{m,n}\}^{+\infty}_{m,n=0}\) in a closed subspace \(Y_ 0\subset Y\) is said to be proper if, for any sequence \(\{a_{m,n}\}^{+\infty}_{m,n=0}\subset\mathbb{C}\), \(\sum^{+\infty}_{m,n=0}a_{m,n}\cdot f_{m,n}\) converges in \(Y\Longleftrightarrow\sum^{+\infty}_{m,n=0}a_{m,n}\cdot\delta_{m,n }\) converges in \(Y\). One of the main results of the paper is the following. Theorem 2.5: A basis \(\{f_{m,n}\}^{+\infty}_{m,n=0}\) in a closed subspace \(Y_ 0\) of \(Y\) is proper if and only if the following conditions are satisfied: (i) For each \(\delta>0\) there exist \(K=K(\delta)\) and \(\varepsilon_ 1=\varepsilon_ 1(\delta)>0\) such that \[ \| f_{m,n};\rho_ 1+\delta,\rho_ 2+\delta\|\leq K\lambda_ m^{\lambda_ m\cdot(\rho_ 1+\varepsilon_ 1)^{-1}}\mu_ n^{\mu_ n\cdot(\rho_ 2 +\varepsilon_ 1)^{-1}},\quad m,n\geq 0. \] (ii) For \(\eta>0\) and all sufficiently small \(\delta=\delta(\eta)>0\) we have \(\| f_{m,n};\rho_ 1+\delta,\rho_ 2+\delta\|\geq\lambda_ m^{\lambda_ m\cdot(\rho_ 1+\eta)^{-1}}\mu_ n^{\mu_ n\cdot(\rho_ 2+\eta)^{-1}}\), where \(m+n\) is sufficiently large and depends on \(\delta\).