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A sheaf of Boehmians. (English) Zbl 1275.46027

The authors have proved that Boehmians on \(n\)-dimensional Euclidean spaces constitute a sheaf (a mathematical structure that is used to organize local information over open sets in a topological space) which may widen the understanding of the theory of Boehmians over locally compact spaces. Theorems and lemmas assist in proving a gluing property of Boehmians over two open sets. The gluing property on Boehmians is then extended to the gluing property of sheaves (see [J. A. Seebach jun. et al., Am. Math. Mon. 77, 681–703 (1970; Zbl 0202.22703)]). It is proved that the Boehmians constitute a sheaf. Through lemmas and a theorem, in Section 2, we have relevant concepts and notations employed in the text. Given further is the description of the construction of Boehmians on open subsets of \(n\)-dimensional Euclidean spaces, which are defined as equivalence classes of fundamental sequences of continuous functions. It is remarked that Boehmians on a closed set are not defined. In order to describe the gluing property, in Section 4, the restriction of a Boehmian to an open set is employed and some lemmas are proved. Theorem 4.9 defines Boehmians as a sheaf and it is then invoked in Section 5 to show that Boehmians are a sheaf. For some set to be considered a sheaf over a topological space, it must be a presheaf, which has an explicit description in Theorem 5.1. The conditions for being a sheaf (see the above quoted paper [loc. cit.]) are shown to be satisfied by Boehmians. Fundamental and Cauchy sequences are discussed in Section 6.

MSC:

46F99 Distributions, generalized functions, distribution spaces
44A40 Calculus of Mikusiński and other operational calculi
44A35 Convolution as an integral transform
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)

Citations:

Zbl 0202.22703
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