Generalized Hyperfunctions on the Circle

Abstract

We give an embedding of the space B(T) of hyperfunctions on the unit circle T in a differential algebra H(T) whose elements are called generalized hyperfunctions. This allows us to define the product of two hyperfunctions without any restriction. We also define pointvalues of a hyperfunction: these pointvalues are elements of an algebra C whose set of invertible elements is denoted C*. In Section 2 we recall and make precise some basic results on classical spaces of functions on T. Section 3 is devoted to our main results: we characterize the set H*(T) of invertible elements of H(T), and, since a generalized hyperfunction may vanish at all classical points without being zero, we give a vanishing theorem. We conclude our work with the study of the Cauchy problem: u′+fu+gu2=0;u(z0)=μ, where f,g∈H(T),z0∈T, and μ∈C*, by giving an existence theorem for a solution u∈H*(T).