Funciones no-estándar y teoría de distribuciones

Abstract

Any sequence of real functions (fn) induces a function f = gen (fn) in the non-standard reals R* by taking the ultraproduct πn and lt; R,fn and gt;/F (F a non-principal ultrafilter). This papeP studies the algebra and the caculus of the functions so obtained. Derivatives and integrals are defined as the obvious non-standard extensions of the corresponding real operators. Continuity instead, is defined via the pseudometric induced by R in R* (f is continuos in ∝ ϵ R* if f(∝ + ε) ≈ f (∝) for infinitesimal ε) . Finally, it is shown that any Schwartz distribution T is represented by a non-standard function f of this kind, in the sense that for any test function g [Formula Matemática]; where g is the canonical extension of g to R*.