Real Analytic Bergman Spaces

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.