A generalized Fourier transformation for L₁(G)-Modules

Abstract

Let G be a compact abelian group with dual G and let Kbe a Banach L,(G)-module. We introduce the notion of character convolution transformation of K which reduces to ordinary Fourier or Fourier- Stieltjes transformation when K is one of the spaces L (G), M(G).We show that the question of what maps G -» K extend to multipliers of K is a question of asking for descriptions of the character convolution transforms. In this setting some results of Helson-Edward and Schoenberg-Eberlein find generalizations, as do some classical results, including the inversion formula and the Parseval relation. We then apply these results to transformation groups, obtaining a variant of a theorem of Bochner and an extension of a theorem of Ryan.