Multiplication of periodic hyperfunctions via harmonic regularization and applications

We build a locally convex algebra of Gevrey type functions defined in the Poincare half-plane in which a class of periodic hyperfunctions on the real line is topologically embedded. This is accomplished via a harmonic regularization method. In this algebra we can give a sense to differential problems involving products of distributions or hyperfunctions which are a priori not defined in the classical setting. Some examples and an application are given.