Representing Systems of Exponentials in Weight Subspaces $H(D)$

In this paper, weight subspaces of the space of analytic functions on a bounded convex domain of the complex plane are considered. Descriptions of spaces that are strongly conjugate to inductive and projective limits of uniformly weight spaces of analytic functions in a bounded convex domain $D\subset \mathbb C$ are obtained in terms of the Fourier–Laplace transformation. For each normed uniformly weight space $H(D,u)$, the smallest linear space $\mathcal H_i(D,u)$ that contains $H(D,u)$ and is invariant under differentiation and the largest linear space $\mathcal H_p(D,u)$ that is contained in $H(D,u)$ and is invariant under differentiation are constructed. Natural locally convex topologies are introduced on these spaces and a description of strongly conjugate spaces in terms of the Fourier–Laplace transformation is presented. The existence of representing exponential systems in the space $\mathcal H_i(D,u)$ is proved.