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Erschienen in:
2012 | OriginalPaper | Buchkapitel
3. Extrafunctions
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Abstract
In this chapter, we define and study different types of extrafunctions. It would be natural to speak of hyperfunctions instead of extrafunctions as mappings of hypernumbers. However, the term hyperfunction is already used in mathematics. So, we call mappings of hypernumbers by the name extrafunction. The main emphasis here is on general extrafunctions and norm-based extrafunctions, which include conventional distributions, hyperdistributions, restricted pointwise extrafunctions, and compactwise extrafunctions, which have studied before in different publications. In Sect. 3.1, the main constructions are described and their basic properties are explicated. For instance, a criterion is found (Theorem 3.1.1) for existence of an extension of the conventional functions to general extrafunctions. Relations between norm-based extrafunctions, distributions, hyperdistributions, pointwise extrafunctions, and compactwise extrafunctions are established. In Sect. 3.2, various algebraic properties of norm-based extrafunctions are obtained.