Integration and Microlocal Analysis in Colombeau Algebras of Generalized Functions

Abstract

We study integration and Fourier transform in the Colombeau algebra Gτ of tempered generalized functions using a general damping factor. This unifies different settings described earlier by J. F. Colombeau, M. Nedeljikov, S. Pilipović, and M. Damsma (for a simplified version). Further we prove characterizations of regularity for generalized functions in two situations: compactly supported or in the image of S′ inside Gτ. Finally we investigate the notion of wave front set in the Colombeau algebra G(Ω), Ω an open subset of Rn, and show that it is in fact independent of the damping measure used for Fourier transform.