Representation of Complex Functions by Orthogonal Series and Faber Series

As is well known, one of the most important methods of representing functions defined on real or complex domains with the help of simpler functions is the method of series expansions. The theory of convergence for functions defined on complex domains, especially for analytic functions, is considerably simpler than for functions defined on real domains. Since we are generally interested in analytic functions, we shall mainly be concerned with series developments in the space L2(G). The first four sections of this chapter are devoted to this topic. An important element in the space L2(G) is the Bergman kernel function, which is useful for the construction of conformal mappings. We talk about the Bergman kernel function in Section 5. Finally, in Section 6, we present the expansion of functions in Faber polynomials in order to obtain certain theorems on the quality of approximation by polynomials.