Integration and microlocal analysis in Colombeau algebras of generalized functions
Generalized functions in the sense of Colombeau extend classical distribution theory to a consistent and efficient method for solving problems in nonlinear analysis. In this paper integration and Fourier transform in the Colombeau algebra \(\mathcal G_r\) of tempered generalized functions are studied. This unifies different settings described by \textit{J. F. Colombeau} [``Elementary introduction to new generalized functions'', North-Holland Math. Stud. 113, Amsterdam, etc. (1985; Zbl 0584.46024)], \textit{M. Nedeljkov} and \textit{S. Pilopovic} [Publ. Inst. Math., Nouv. Sér. 52, 95-104, 105-112 (1992; Zbl 0790.46024 and Zbl 0790.46025)] and \textit{M. Damsma} [Memorandum No. 954, University Twente (1991)]. Some characterizations of regularity for generalized functions are proved. As well the notion of wave front set in the Colombeau algebra \(\mathcal G(\Omega)\), \(\Omega\) an open set of \(\mathbb{R}^n\), is investigated.
scientific article; zbMATH DE number 1384926