Algebras of generalized functions with smooth parameter dependence

Abstract

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring Ksm of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of Ksm and establish some properties of its ideals.

Categorical frameworks for generalized functions

“…We will start with Elements (u ε ) of C ∞ ( ) I are arbitrary nets, indexed in ε ∈ I , of smooth functions on . There are studies of Colombeau-like algebras with smooth or continuous ε-dependence (see [7,22] and references therein). In [21], it has been proved that a very large class of equations have no solution if we request continuous dependence with respect to ε ∈ I .…”

Asymptotic Ideals (Ideals in the Ring of Colombeau Generalized Constants with Continuous Parametrization)

“…K is an exchange ring [13], i.e., for each a ∈ K, there exists an idempotent e ∈ K such that a + e is invertible. Unlike K, K cnt is not an exchange ring [5,Lemma 4.3]. Like K, K cnt is a Gelfand ring [5, Lemma 4.5], i.e., every prime ideal is contained in a unique maximal ideal.…”

“…Like K, K cnt is a Gelfand ring [5, Lemma 4.5], i.e., every prime ideal is contained in a unique maximal ideal. Like K, K cnt is a Bezout ring [5,Prop. 4.26], i.e., every finitely generated ideal is principal.…”

“…4.26], i.e., every finitely generated ideal is principal. Like R, R cnt is an l-ring (or lattice-ordered ring) [5,Prop. 4.13].…”

“…Apart from the obvious interest of such a study to asymptotic analysis, such equivalence classes of functions also naturally arise in generalized function theory as the ring of generalized constants K cnt of the algebra of Colombeau generalized functions (see §2). The ring K cnt of generalized constants with continuous dependence on the parameter has been introduced and studied in [5], where it is also shown that this ring is isomorphic to the ring of generalized constants with smooth dependence. In fact, the study of the ring K cnt amounts to the study of the asymptotics at zero of moderate continuous functions on (0, 1].…”