Complete Pick Spaces: Theory and Examples

This thesis explores the Nevanlinna Pick problem in complex analysis, which involves finding an analytic function that interpolates given data points within the complex unit disc. It focuses on reproducing kernel Hilbert spaces (RKHS) known as Pick spaces, which possess the analog of the interpolation property. The main results of the thesis relate to the study of a concrete and simple RKHS of functions on the real line. This space is completely characterized, with proofs of the Pick property, study of interpolating sequences, multipliers, Carleson measures, invariant subspaces, and the solution to the Corona problem.