**Abstract** The usual theory of Bergman spaces begins with a domain $D$ with weight $\omega$.One takes the elements of $L^p(D)$ which are holomorphic. One could also take the real parts of such functions and study them. Two characteristic properties of Bergman spaces are that all functions are real analytic, and bounded point evaluation holds. These are well known examples. In this talk we consider much larger closed subspaces of functions in $L^p(\mathbb{C},\omega)$ where all functions are real analytic, and bounded point evaluation holds. The functions are expressed as power series in $z$ and $z g(z)$, where $g(z)$ is an entire function whose zeros spread out toward infinityin a certain way. This is a mild condition which is easy to satisfy. For instance, $g(z)= \frac{\sin z\, \sin \tau z\, \sin \tau^2 z}{z^2}$ works, where $\tau^3=1,\ \tau\neq 1$. The proof involves CR wedge extension.