Introduction to the more known spaces of holomorphic functions in the disc and in a half plane. Analysis of their properties, with attention to the proofs techniques. The focus is on Hardy and (weighted) Bergman spaces on the disc and in a half plane, Paley-Wiener spaces and Bernstein spaces.
Knowledge of the topics and results, and application to exercises that need also computational techniques.
Hardy spaces Hp(D) on the unit disc. Function spaces with reproducing kernel. Bergman spaces Ap(D) and weighted Bergman spaces Aνp(D). Lp boundedness of the Bergman and Cauchy-Szego projections. Fourier tranform on R of the spaces L1 and L2. Paley—Wiener theorems. Bergman and Hardy spaces in the upper half plane. Introduction to several complex variables theory. Holomorphic fuctions on the unit disc, Hardy and Bergman spaces, projections and boundedness in Lp.