For a domain U in C^n, let A^p(U) denote the subset of functions belonging to L^p(U) that are holomorphic. Given a map f in L^p(U), can we use it to construct a "natural" function in A^p(U)? The Bergman kernel is a useful tool that can be used to investigate this question, but there are certain limitations. I will show that Hilbert space methods may still be employed to attack this problem, even when the Bergman kernel fails to do the job. I plan to focus on a class of model domains and introduce new family of integral kernels which avoid the issues that limit the Bergman kernel. This work is joint with Debraj Chakrabarti and Jeff McNeal. Speaker(s): Luke Edholm (U(M))