Convergence with Respect to Admissible Sequences of Weights for a Holomorphic Space of a Sequence of Hilbert-Valued Rational Functions

Abstract

Let $D \subset\C^N$ be a domain, $ H $ be a locally convex space with the topology defined by a sequence of Hilbert semi-norms. Denote $\mathscr H(D,H)$ the space of $ H $-valued holomorphic functions on $D$ and $\mathscr R(\C^N,H)$ the space of $ H $-valued rational functions on $\C^N.$ In this paper, we set up terminology of \textit{an admissible sequence $ \mathcal W = \{w_m\}_{m\ge1}$ of weights for $\mathscr H(D, F)$} and study
sufficient conditions on $ \mathcal W$ to ensure convergence of $w_m(\|r_m-f\|_m^2)$ (pointwise/ in capacity/ uniformly on compact subsets) to $0$ on the whole domain $D$ except for a pluripolar subset provided the pointwise (rapid) $\mathcal W$-convergence of $\{r_m-f\}_{m \ge 1}$ to $0$ occurs on a Borel non-pluripolar subset $ X $ for every $f\in \mathscr H(D,H)$ and $\{r_m\}_{m \ge 1} \subset \mathscr R(\C^N,H),$
in both of cases $X$ lies in $D,$ and $X$ lies in the boundary $ \partial D $ of a bounded domain $ D.$