Generalization of Hp functions in tubes. I

Abstract

A generalization of the Hardy Hp functions corresponding to tubes in and of other related spaces of functions is given. The holomorphic functions in our new spaces are represented as Fourier-Lapiace integrals and properties of the representing functions are obtained. In certain cases the holomorphic functions have distributional boundary values in the strong topology of the tempered distributions, and when the distributional boundary values are of a certain type the holomorphic functions can be represented as the Cauchy and Poisson integrals of the boundary values. All theorems have converse results. The corresponding results for the Hp and previously known related spaces are special cases of the theorems presented here.