Necessary condition of fundamental principle for invariant subspaces on unbounded convex domain

In this paper we study the spaces $H(D)$ of analytic functions in convex domains of the complex plane as well as subspaces $W(\Lambda,D)$ of such spaces. A subspace $W(\Lambda,D)$ is the closure in the space $H(D)$ of the linear span of the system $\mathcal{E}(\Lambda)=\{z^n \exp(\lambda_k z)\}_{k=1,n=0}^{\infty,n_k-1}$, where $\Lambda$ is the sequence of different complex numbers $\lambda_k$ and their multiplicities $n_k$. This subspace is invariant with respect to the differentiation operator. The main problem in the theory of invariant subspaces is to represent all its functions by using the eigenfunctions and associated functions of the differentiation operator, $z^n e^{\lambda_k z}$. In this paper we study the problem of the fundamental principle for an invariant subspace $W(\Lambda,D)$, that is, the problem of representing all its elements by using a series constructed over the system $\mathcal{E}(\Lambda)$. We obtain simple geometric conditions, which are necessary for the existence of a fundamental principle. These conditions are formulated in terms of the length of the arc of the convex domain and the maximum density of the exponent sequence.