The Generalized Stieltjes and Fourier Transforms of Certain Spaces of Generalized Functions

Abstract

Inspired of Roumieu and Beurling definitions, we defined spaces of ultradifferentiable functions of rapid descents in L^p spaces which shown to be closed with respect to the classical Stieltjes transform. The generalized Stieltjes transform is thereby defined on the duals through the generalization of the Parseval’s equation. Further, we obtained spaces and derived certain related results justifying multiplication. Adopting the concept of Boehmian spaces, the celebrated Fourier transforms were extended to a so-called space of ultradifferentiable Boehmians.

Key words and phrases. Generalized Function, Ultradistribution, Tempered Distribution, Stieltjes Transform, Roumieu Ultradistribution.

2000 Mathematics Subject Classification. 40H05, 46A45