Sampling measure on Bergman spaces

Abstract

In this paper, we investigate the relationship between sampling measure μ \mu , Berezin transform μ ~ \tilde {\mu } and r r -averaging transform μ ^ r \widehat {\mu }_r on Bergman spaces. Compared with some results of Luecking [Amer. J. Math. 107 (1985), pp. 85–111], our results provide an equivalent description of sampling measures, which reveals the reason why 1 / μ ^ r ∈ L ∞ 1/\hat {\mu }_r\in L^\infty or 1 / μ ~ ∈ L ∞ 1/\tilde {\mu }\in L^\infty does not make sure that μ \mu is a sampling measure on Bergman spaces.