Egorov generalized function algebra
Given an open subset of
, Yu.V. Egorov [a1] defined the generalized function algebra
as the factor algebra of
modulo the ideal of sequences
which vanish eventually on every compact subset of
. The family
provides a sheaf of differential algebras on
. Convolution with a sequence of mollifiers
, where
converges to the Dirac measure, gives an imbedding of the space
of compactly supported distributions into
which respects derivatives as well as supports. It can be extended as a sheaf morphism to an imbedding of the space of distributions
.
As a generalized function algebra, can be employed to study non-linear partial differential equations. In particular, Egorov has used the algebra to construct generalized solutions to boundary value problems as well as evolution equations. In the latter case the spatial derivative
may be replaced by the difference operator
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where denotes the
th unit vector in
. This way partial differential equations are approximated by ordinary difference-differential equations in the algebra
.
See also Generalized function algebras.
References
[a1] | Yu.V. Egorov, "A contribution to the theory of generalized functions" Russian Math. Surveys , 45 : 5 (1990) pp. 1–49 |
Egorov generalized function algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Egorov_generalized_function_algebra&oldid=17911