Unifying order structures for Colombeau algebras

Abstract

We define a general notion of set of indices which, using concepts from pre‐ordered sets theory, permits to unify the presentation of several Colombeau‐type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau's notion of big‐O such that its usual properties continue to hold. Using this generalized notion of big‐O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.

The category of Colombeau algebras

“…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 .…”

“…In [13], the general notion of sets of indices has been introduced. For reader's convenience, in this section we recall the notations and notions from [13] that we will use in the present work.…”

“…For reader's convenience, in this section we recall the notations and notions from [13] that we will use in the present work. For all the proofs, we refer to [13].…”

“…, r ∈ R >0 and x ∈ R n , we use the symbol r ϕ to denote the function x ∈ R n → 1 r n · ϕ( x r ) ∈ R, see [13]. With the usual notations of [5], we define I e := A 0 , I e := {A q | q ∈ N}, and for ε, e ∈ I e , we set ε ≤ e iff there exists r ∈ R >0 such that r ≤ 1 and ε = r e. Then I e := (I e , ≤, I e ) is a set of indices used in this framework to unify and simplify the full algebra G e (see [13,Sect.…”

Asymptotic gauges: Generalization of Colombeau type algebras

“…Set of indices : The results of this paper are proved for a generic set of indices, a new unifying structure introduced in . This permits to include several Colombeau algebras in the same framework and notations: the special algebra , the full algebra and the nonstandard analysis based algebra of asymptotic functions .…”