  The usual examples of Bergman spaces consist of the closure of an algebra of
holomorphic functions on a domain. One can also take the real part of such
functions, but essentially one is looking at the same object. In this paper the
author shows that the properties of real analyticity and bounded point
evaluation can be preserved under closure in a weighted $L^p$ space, if one
takes the algebra generated by $z$ and $(\bar z)g(z)$, where $g$ is an entire
function satisfying some condition on the distribution of zeros near $\infty$.
The condition is not difficult to satisfy with simple examples. The resulting
spaces are much larger than the ordinary Bergman spaces. The main example is
with a Gaussian weight like the Fock space. One can get good approximation of
the usual Fock space by a sequence of these real analytic Bergman spaces.
