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A global theory of algebras of generalised functions

A global theory of algebras of generalised functions
A global theory of algebras of generalised functions
We present a geometric approach to defining an algebra (M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space ?(M) of distributions on M.
Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of (M). (M) is a differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ?(M) into (M) that renders ?(M) a faithful subalgebra of (M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus (M ) retains all the distinguishing properties of the local theory in a global context.
algebras of generalized functions, colombeau algebras, distributions on manifolds, calculus on infinite-dimensional spaces
0001-8708
50-72
Grosser, M.
ef6421c5-8bbb-4734-9a3d-e4b4ece9c33a
Kunzinger, M.
553c5264-8f37-4e7c-8316-09e7562fbbf7
Steinbauer, R.
03564261-185d-4e1d-97ed-a7633d552a34
Vickers, J. A.
719cd73f-c462-417d-a341-0b042db88634
Grosser, M.
ef6421c5-8bbb-4734-9a3d-e4b4ece9c33a
Kunzinger, M.
553c5264-8f37-4e7c-8316-09e7562fbbf7
Steinbauer, R.
03564261-185d-4e1d-97ed-a7633d552a34
Vickers, J. A.
719cd73f-c462-417d-a341-0b042db88634

Grosser, M., Kunzinger, M., Steinbauer, R. and Vickers, J. A. (2002) A global theory of algebras of generalised functions. Advances in Mathematics, 166 (1), 50-72. (doi:10.1006/aima.2001.2018).

Record type: Article

Abstract

We present a geometric approach to defining an algebra (M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space ?(M) of distributions on M.
Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of (M). (M) is a differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ?(M) into (M) that renders ?(M) a faithful subalgebra of (M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus (M ) retains all the distinguishing properties of the local theory in a global context.

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More information

Submitted date: 14 April 2000
Published date: 2002
Keywords: algebras of generalized functions, colombeau algebras, distributions on manifolds, calculus on infinite-dimensional spaces

Identifiers

Local EPrints ID: 39007
URI: https://eprints.soton.ac.uk/id/eprint/39007
ISSN: 0001-8708
PURE UUID: fa832b7b-e70c-43ab-bdd0-f0426bca41aa
ORCID for J. A. Vickers: ORCID iD orcid.org/0000-0002-1531-6273

Catalogue record

Date deposited: 19 Jun 2006
Last modified: 08 Oct 2019 00:57

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