Interpolation and sampling in small Bergman spaces

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 $(1-\mathrm{\mid }z\mathrm{\mid }{)}^{-\alpha }$, $0<\alpha <1$. These results make the Hardy space $H^2$ appear naturally as a “degenerate” endpoint case for the class of Bergman spaces under study.