Product of Hyperfunctions with Disjoint Support

Abstract

We prove that if two hyperfunctions on the unit circle have disjoint support, then the convolution of their Fourier coefficients multiplied with a weight is zero when the weight goes to 1. We prove this by using the Fourier-Borel transform and the G-transform of analytic functionals. The proof is inspired by an article by Yngve Domar. In the end of his article he proves the existence of a translation-invariant subspace of a certain weighted l^p-space. This proof has similarities to our proof, so we compare them. We also look at other topics related to Domar's article, for example the existence of entire functions of order less than or equal to 1 under certain restrictions on the axes. We will see how the Beurling-Malliavin theorem gives some answers to this question. Finally, we prove that if two hyperfunctions on the real line have compact and disjoint support, then the convolution of their Fourier transforms multiplied with a weight is zero when the weight goes to 1.