Abstract
Ultrafunctions are a particular class of functions defined on a non-Archimedean field $${\mathbb{R}^{\ast } \supset \mathbb{R}}$$ . They have been introduced and studied in some previous works (Benci, Adv Nonlinear Stud 13:461–486, 2013; Benci and Luperi Baglini, EJDE, Conf 21:11–21, 2014; Benci, Basic Properties of ultrafunctions, to appear in the WNDE2012 Conference Proceedings, arXiv:1302.7156 , 2014). In this paper we introduce a modified notion of ultrafunction and discuss systematically the properties that this modification allows. In particular, we will concentrate on the definition and the properties of the operators of derivation and integration of ultrafunctions.
Highlights
In some recent papers the notion of ultrafunction has been introduced and studied [1,8,9]
Ultrafunctions are a particular class of functions defined on a non-Archimedean field R∗ ⊃ R
As we showed in our previous works, when working with ultrafunctions we associate to any continuous function f : RN → R an ultrafunction f : (R∗)N → R∗ which extends f ; more exactly, to any vector space of functions V ( ) ⊆ L2( ) ∩ C( ) we associate a space of ultrafunctions V ( )
Summary
In some recent papers the notion of ultrafunction has been introduced and studied [1,8,9]. One property that is missing, in general, is the “locality”: the extensions of operators that are local on V ( ) may not be local on V ( ). This problem is related to the properties of a particular basis of the spaces of ultrafunctions, called “Delta basis” (see [8,9]). The techniques on which the notion of ultrafunction is based are related to non-Archimedean mathematics (NAM) and to nonstandard analysis (NSA). -limit (see [1,8,9]) In this paper this notion will be considered known; for sake of completeness, we will recall its basic properties in the Appendix
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