Extremal problems and bergman projections

Abstract

Studying extremal problems on Bergman spaces is rather new and techniques used are usually specific to the problem to be solved. However, a 2014 paper by T. Ferguson developed a systematic method using Bergman projections for solving extremal problems on Bergman spaces Ap on the unit disc with 1 < p < 1. We extended this method to weighted Bergman spaces Ap and in some special cases, to extremal problems defined by linear functionals of evaluations at points in the disc other than the origin. We computed the kernels of several such functionals. We also computed the Bergman projections of some functions related to these kernels. Using these projections, we solved a few extremal problems explicitly. Our results have the potential to be extended to the case p = 1 and to more general spaces. We also gave a proof of the existence of the solutions to the extremal problems on the Bergman spaces A1 defined by functionals with kernels that extend to the closed disc continuously.