Asymptotic gauges: generalization of Colombeau type algebras (Q2792253)

The authors use the general notion of a set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on a more general ``growth condition'' formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based on Colombeau-type algebras in a unique framework. They compare Colombeau algebras generated by asymptotic gauges with other analogous constructions, and they study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. At the end of the paper, it is proven that, in the newly constructed framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This property marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restrictions on the coefficients can be solved.