Representing systems of exponentials in projective limits of weighted subspaces of $H(D)$

We consider uniformly weighted spaces of analytic functions on a boundedconvex domain in the complex plane with convex weights. For every uniformlyweighted normed space $H(D,\varphi)$ we define a special inductive limit$\mathcal H_i(D,\varphi)$ of normed spaces and a special projective limit$\mathcal H_p(D,\varphi)$ of normed spaces. We prove that$\mathcal H_i(D,\varphi)$ is the smallest locally convex space whichcontains $H(D,\varphi)$ and is invariant under differentiation, and$\mathcal H_p(D,\varphi)$ is the largest such space which is contained in $H(D,\varphi)$. We construct a representing system of exponentials in the projective limit$\mathcal H_p(D, \varphi)$ and estimate the redundancy of this system.

analytic functions, weighted spaces, locally convex spaces, sufficient sets, representing systems of exponentials