Representation of functions in locally convex subspaces of $A^\infty (D)$ by series of exponentials

Abstract

Let $D$ be a bounded convex domain in the complex plane, $\mathcal M_0=(M_n)_{n=1}^\infty$ be a convex sequence of positive numbers satisfying the “non-quasi-analyticity” condition: $$\sum_n\frac {M_n}{M_{n+1}}<\infty,$$ $\mathcal M_k=(M_{n+k})_{n=1}^\infty$, $k=0,1,2,3,\ldots$ be the sequences obtained from the initial ones by removing first $k$ terms. For each sequence $\mathcal M_0=(M_n)_{n=1}^\infty$ we consider the Banach space $H(\mathcal M_0,D)$ of functions analytic in a bounded convex domain $D$ with the norm: $$\|f\| ^2=\sup_n \frac 1{M_n^2}\sup_{z\in D}|f^{(n)}(z)|^2.$$ In the work we study locally convex subspaces in the space of analytic functions in $D$ infinitely differentiable in $\overline D$ obtained as the inductive limit of the spaces $H(\mathcal M_k,D)$. We prove that for each convex domain there exists a system of exponentials $e^{\lambda_nz}$, $n\in \mathbb{N}$, such that each function in the inductive limit $f\in \lim {\text ind}\, H(\mathcal M_k,D):=\mathcal H(\mathcal M_0,D)$ is represented as the series over this system of exponentials and the series converges in the topology of $\mathcal H(\mathcal M_0,D)$. The main tool for constructing the systems of exponentials is entire functions with a prescribed asymptotic behavior. The characteristic functions $L$ with more sharp asymptotic estimates allow us to represent analytic functions by means of the series of the exponentials in the spaces with a finer topology. In the work we construct entire functions with gentle asymptotic estimates. In addition, we obtain lower bounds for the derivatives of these functions at zeroes.