Colombeau generalized function algebras
Let be an open subset of
, and let
be the algebra of compactly supported smooth functions. In the original definition, J.F. Colombeau [a2] started from the space
of infinitely Silva-differentiable mappings from
into
. The space of distributions
is just the subspace of linear mappings
. Let
![]() |
and let
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The subalgebra is defined by those members
such that for all compact subsets
and for all multi-indices
there is an
such that for all
, the supremum of
over
is of order
as
. The ideal
is defined by those members
such that for all compact subsets
and all
there is an
such that for all
and
, the supremum of
over
is of order
as
. The Colombeau generalized function algebra is the factor algebra
. It contains the space of distributions
with derivatives faithfully extended (cf. also Generalized function, derivative of a). The asymptotic decay property expressed in
together with an argument using Taylor expansion shows that
is a faithful subalgebra.
Later, Colombeau [a3], [a4] replaced the construction by a reduced power of with index set
: Let
be the algebra of all nets
such that for all compact subsets
and all multi-indices
there is an
such that the supremum of
over
is of order
as
(cf. also Net (directed set)). Let
be the ideal therein given by those
such that for all compact subsets
, all
and all
, the supremum of
over
is of order
as
. Then set
. There exist versions with the infinite-order Sobolev space
in the place of
,
, or with other topological algebras.
It is possible to enlarge the class of mollifiers (hence the index set in the reduced power construction) to produce a version for which smooth coordinate changes commute with the imbedding of distributions. This way Colombeau generalized functions can be defined intrinsically on manifolds. Generalized stochastic processes with paths in
have been introduced as well.
The subalgebra is defined by interchanging quantifiers: For all compact sets
there is an
such that for all
, the supremum of
on
is of order
as
. One has that
, and
plays the same role in regularity theory here as
does in distribution theory (for example,
and
implies
, where
denotes the Laplace operator).
For applications in a variety of fields of non-linear analysis and physics, see [a1], [a4], [a5], [a6], [a7].
See also Generalized function algebras.
References
[a1] | H.A. Biagioni, "A nonlinear theory of generalized functions" , Springer (1990) |
[a2] | J.F. Colombeau, "New generalized functions and multiplication of distributions" , North-Holland (1984) |
[a3] | J.F. Colombeau, "Elementary introduction to new generalized functions" , North-Holland (1985) |
[a4] | J.F. Colombeau, "Multiplication of distributions. A tool in mathematics, numerical engineering and theoretical physics" , Springer (1992) |
[a5] | "Nonlinear theory of generalized functions" M. Grosser (ed.) G. Hörmann (ed.) M. Kunzinger (ed.) M. Oberguggenberger (ed.) , Chapman and Hall/CRC (1999) |
[a6] | M. Nedeljkov, S. Pilipović, D. Scarpalézos, "The linear theory of Colombeau generalized functions" , Longman (1998) |
[a7] | M. Oberguggenberger, "Multiplication of distributions and applications to partial differential equations" , Longman (1992) |
Colombeau generalized function algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Colombeau_generalized_function_algebras&oldid=50118