A Convenient Notion of Compact Set for Generalized Functions. Issue 1 (3rd April 2017)

• Record Type: Journal Article
• Authors: Giordano, Paolo; Kunzinger, Michael
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 61:Issue 1(2018)
• Page Start: 57
• Page End: 92
• Publication Date: 2017-04-03
• Subjects: functionally compact sets, -- Colombeau generalized functions, -- generalized smooth functions, -- locally convex modules; Primary 46F30, -- 46A13, -- 13J99; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091516000559
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract : We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.