Asymptotic gauges: Generalization of Colombeau type algebras

Abstract

Abstract. We use the general notion of 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 more general "growth condition" formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our 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 marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restriction on the coefficients can be solved.

The category of Colombeau algebras

“…(see also [5,12]). A step towards the analysis of these problems is the generalization of the role of the infinite nets (ε −n ) ∈ R (0,1] , n ∈ N, appearing in the definition of Colombeau algebras, see (1.1).…”

“…A step towards the analysis of these problems is the generalization of the role of the infinite nets (ε −n ) ∈ R (0,1] , n ∈ N, appearing in the definition of Colombeau algebras, see (1.1). An understanding of the role of the nets (ε −n ) ∈ R (0, 1] has already been realized through the notions of asymptotic scales, (C, E, P)-algebras, (M, N , V P )-algebras, exponent weights and asymptotic gauges, see [6][7][8][9]12,14,15] and references therein.…”

“…In fact, Colombeau AG-algebras include in the same abstract framework all known Colombeau-like algebras, like the special one G s , the full one G e , [5], the NSA based algebra of asymptotic functionsĜ, [23], the diffeomorphism invariant algebras G d , G 2 andĜ, [5], the Egorov algebra, [10,11,25], and the algebra of non-standard smooth functions * C ∞ ( ), [4,18]; see [12,13] and below. The simplicity of the approach with AGs lies, for all the algebras mentioned above, in the use of the simple logical structure of quantifiers that characterizes the special algebra G s .…”

“…We shall frequently use (1.2) as a paradigmatic motivation, even if the solution of this trivial ODE is not one of the aim of the present paper. For the solution and uniqueness of all R-linear ODE in a minimal Colombeau AG-algebra, see [12].…”

“…In [12], we introduced the notion of asymptotic gauge. The idea was to use it as an asymptotic scale that generalizes the role of the polynomial family (ε n ) ε∈(0,1],n∈N in classical constructions of Colombeau algebras.…”