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New versions of the Colombeau algebras. (English) Zbl 1115.46035
This paper constitutes an important contribution in the growing field of theories of nonlinear generalized functions. Among other results, a new version of the simplified (or special) algebra of J.–F. Colombeau is constructed. This new algebra is a factor space of moderate elements (formed by sequences of families of smooth functions), modulo negligible ones. Let us remark that this construction can be reinterpreted as bi-parametric $$(C,E,P)$$-algebra, as they were introduced by J.–A. Marti. The construction presented here shares the main properties of the Colombeau algebra. In addition, representation of generalized functions exists as weak asymptotic series whose coefficients are distributions, which is a new feature (albeit in the stream of previous works of the same author).
The author, furthermore, gives a construction of Colombeau type algebras generated by harmonic or polyharmonic regularizations of distributions connected with the half plane, the latter being, as far as the reviewer knows, a true novelty. In the same spirit, the author introduces Colombeau type algebras based on analytic representation of distributions, connected with an octant.
Some applications are given, such as finding asymptotic expressions for the product of distributions (for example, for $$\delta^2$$, as already given earlier by Li Bang He and Li Ya–Qing). More interestingly, the approach given here seems to be very efficient for solving nonlinear hyperbolic systems, with the introduction, for example, of $$\delta$$-shock wave solutions.

##### MSC:
 46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) 46F05 Topological linear spaces of test functions, distributions and ultradistributions 46F10 Operations with distributions and generalized functions
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