  Transposing the Berezin quantization into the setting of analytic microlocal
analysis, we construct approximate semiclassical Bergman projections on
weighted $L^2$ spaces with analytic weights, and show that their kernel
functions admit an asymptotic expansion in the class of analytic symbols. As a
corollary, we obtain new estimates for asymptotic expansions of the Bergman
kernel on $\mathbb{C}^n$ and for high powers of ample holomorphic line bundles
over compact complex manifolds.
