Interpolation and sampling in small Bergman spaces

TL;DR

This paper investigates Carleson measures, interpolating, and sampling sequences in weighted Bergman spaces with rapidly growing radial weights, revealing how Hardy space $H^2$ emerges as a special endpoint case.

Contribution

It extends the understanding of sampling and interpolation in weighted Bergman spaces with faster-than-standard weights, connecting these spaces to Hardy space as a limit case.

Findings

Characterization of Carleson measures for these spaces

Description of interpolating sequences in the weighted Bergman spaces

Analysis of sampling sequences and their properties

Abstract

Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-\alpha}$, $0<\alpha<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study.