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

K. P. Isaev ; K. V. Trounov ; R. S. Yulmukhametov

Izvestiya. Mathematics , Tome 83 (2019) no. 2, pp. 232-250.

Résumé

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.

Détail

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

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