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A Global Theory of Algebras of Generalized Functions

✍ Scribed by M. Grosser; M. Kunzinger; R. Steinbauer; J.A. Vickers

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
206 KB
Volume
166
Category
Article
ISSN
0001-8708

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✦ Synopsis

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

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