Harmonic Bergman spaces

In this study, firstly some fundamental properties of harmonic functions andharmonic polynomials are presented, and then the structure of harmonic Bergmanspaces are investigated. Several properties of these spaces are analogues to those ofthe Bergman spaces of analytic functions. Most importantly, these spaces possessreproducing kernels and these kernels induce a projection (called Bergman projec-tion) from Lp spaces into these spaces. These reproducing kernels are investigatedin detail.In this thesis, which consists of six chapters, firstly basic properties of harmonicfunctions, are given. The properties of bounded harmonic functions and positiveharmonic functions are investigated in chapters three and four, respectively. In thefifth chapter, the properties of harmonic polynomials are given in three sections.Firstly, a decomposition theorem for homogeneous polynomials is given and thenL2(S) space is written as a direct sum of spaces of homogeneous harmonic poly-nomials. This chapter is terminated by investigations of zonal harmonics. Thelast chapter, starting with properties of harmonic Bergman spaces,is terminated byderivation of an explicit formula for the Bergman reproducing kernel on the unit ball.