Hilbert Transforms

Abstract

Because the convolution of two hyperfunctions is an established concept, we may use it to define the Hilbert transform of a hyperfunction. On the other hand, the classical Hilbert transform is also linked to the Cauchy-type integral. These two parents have unfortunately led to the fact that there is no adopted standard definition of the Hilbert transform. After some hesitation I have chosen the definition which relays on the Cauchy-type integral.