Generalized Functions Beyond Distributions

TL;DR

This paper introduces a modified class of ultrafunctions defined on non-Archimedean fields, exploring their properties and the behavior of derivation and integration operators within this framework.

Contribution

It presents a new version of ultrafunctions and systematically analyzes their properties, especially focusing on differentiation and integration operators.

Findings

Modified ultrafunctions extend previous concepts.

Properties of derivation and integration are characterized.

Framework facilitates analysis beyond classical distributions.

Abstract

Ultrafunctions are a particular class of functions defined on a Non Archimedean field R^{*}\supset R. They have been introduced and studied in some previous works ([1],[2],[3]). In this paper we introduce a modified notion of ultrafunction and we discuss sistematically the properties that this modification allows. In particular, we will concentrated on the definition and the properties of the operators of derivation and integration of ultrafunctions.