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

We consider uniformly weighted spaces of analytic functions on a bounded convex domain in the complex plane with convex weights. For every uniformly weighted 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 which contains $H(D,\varphi)$ 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.