Colombeau generalized functions: A theory based on harmonic regularization (Q1272299)

\textit{J. F. Colombeau} constructed an associative algebra of new generalized functions [``Elementary introduction to new generalized functions'', Amsterdam (1985; Zbl 0584.46024)]. The new generalized functions and the distributions were related via the notion of an associated distribution or asymptotics. However, the regularization used in the embedding of the space of distributions [\textit{M. Oberguggenburger}, ``Multiplication of distributions and applications to partial differential equations'', New York (1992; Zbl 0818.46036)] leads to serious difficulties in calculating asymptotics and using the theory in applications. In this paper, the author constructs a version of the Colombeau theory based on a polyharmonic regularization for the embedding of the distribution space. This approach permits one to find asymptotic expansions of products of distributions and to use the theory in analytic applications. Earlier, the author has shown [Dokl. Akad. Nauk. SSSR 314, No. 1, 159-164 (1990; Zbl 0743.46028); Mat. Zametki 57, No. 5, 765-783 (1995; Zbl 0868.46028)] that harmonic regularizations are efficient in applications.