Generalized Hyperfunctions on the Circle

We give an embedding of the space BŽ . of hyperfunctions on the unit circle in a differential algebra HŽ . 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 . Section 3 is devoted to our main results: we characterize the set H*Ž . of invertible elements of HŽ ., 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; uŽz0. , where f, g HŽ ., z0 , and C*, by giving an existence theorem for a solution u H*Ž ..