Harmonic Analysis on Hilert Spaces of Holomorphic Functions

Abstract

For the holomorphic functions of one complex variable, say, on the unit disk, the expansion into a series of monomials (power series) is the crucial analytic tool for the study of Hardy or Bergman type spaces. In the multi-variable setting the fine structure of spaces of holomorphic functions is much more complicated, reflecting the geometry of the underlying domain in ℂn. Noting that the classical power series expansion is essentially Fourier analysis on the 1-torus, it is natural to use other Lie group actions in n variables to obtain generalized expansions of holomorphic functions. In general, these Lie groups will be non-abelian so that the standard concepts of Fourier analysis have to be extended to the well-known Peter-Weyl theory. Since we are interested in holomorphic functions, the Fourier analysis has to be combined with a Paley-Wiener type characterization of such functions, and it is here that the geometry of the underlying domains (polar decomposition) plays a crucial role.KeywordsHilbert SpaceHolomorphic FunctionHardy SpaceBergman SpaceJordan AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.