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.


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)


Zbl 0202.22703
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