Colombeau Algebras are differential algebras of generalized functions (that include the space of distributions) that are defined using a quotient set procedure involving particular classes of nets in a basic space E = (C∞(Ω))A, where Ω is an open subset of R n and A is an index set. The choice of such nets depends mainly on their asymptotic behavior over a suitable index set A. Many variants of Colombeau Algebras existing in the literature occur mainly due to different choices of the index set (and to the choice of asymptotic behavior). A purpose of this paper is to formally unify some of these algebras, redefining the asymptotic behavior on an abstract (pre-ordered) set of indices, and generalizing the corresponding “Landau big-O” notion. Such notions are reformulated in order to simplify the definition of diffeomorphism invariant algebra and to generalize some theorems which hold in the case of a special algebra

Francesco Tschinke (2015). MR3377117 Reviewed Giordano, Paolo; Nigsch, Eduard A. Unifying order structures for Colombeau algebras. Math. Nachr. 288 (2015), no. 11-12, 1286–1302. (Reviewer: Francesco Tschinke) [Altro].

MR3377117 Reviewed Giordano, Paolo; Nigsch, Eduard A. Unifying order structures for Colombeau algebras. Math. Nachr. 288 (2015), no. 11-12, 1286–1302. (Reviewer: Francesco Tschinke)

TSCHINKE, Francesco
2015-01-01

Abstract

Colombeau Algebras are differential algebras of generalized functions (that include the space of distributions) that are defined using a quotient set procedure involving particular classes of nets in a basic space E = (C∞(Ω))A, where Ω is an open subset of R n and A is an index set. The choice of such nets depends mainly on their asymptotic behavior over a suitable index set A. Many variants of Colombeau Algebras existing in the literature occur mainly due to different choices of the index set (and to the choice of asymptotic behavior). A purpose of this paper is to formally unify some of these algebras, redefining the asymptotic behavior on an abstract (pre-ordered) set of indices, and generalizing the corresponding “Landau big-O” notion. Such notions are reformulated in order to simplify the definition of diffeomorphism invariant algebra and to generalize some theorems which hold in the case of a special algebra
2015
Francesco Tschinke (2015). MR3377117 Reviewed Giordano, Paolo; Nigsch, Eduard A. Unifying order structures for Colombeau algebras. Math. Nachr. 288 (2015), no. 11-12, 1286–1302. (Reviewer: Francesco Tschinke) [Altro].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/325650
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