A distribution, being a generalized function, is called non-singular if it is (defined by) an actual smooth function (def. below). The subspace of non-singular distributions is a dense subspace of that of all distributions.
(non-singular distributions)
For , a smooth function induces a distribution
by integration of smooth functions against .
This construction defines a linear inclusion
of the real vector space of smooth functions into that of distributions.
The distributions arising this way are called the non-singular distributions.
(singular support of a distribution)
For let be a distribution. Then the singular support is the subset of points such that for every open neighbourhood the restriction is singular, hence not a non-singular distribution (def. ).
(non-singular distributions are dense in all distributions)
The inclusion of non-singular distributions into all distributions
from def. exhibits a dense subspace of (equipped with the dual space topology, this def. or Hörmander 90, p. 38):
Every distribution is the limit of a sequence of smooth functions in that for every compactly supported smooth function we have that the value is the limit of integrals
Analogous statements hold for smaller classes of distributions: Also the inclusion
of compactly supported smooth functions (bump functions) into the space of tempered distributions is dense.
(e.g. Hörmander 90, lemma 7.1.8)
Proposition justifies to think of distributions as “generalized functions”, and to seek generalizations of standard operations on functions, such as concepts of product of distributions and composition of distributions?.
Last revised on November 2, 2017 at 18:11:27. See the history of this page for a list of all contributions to it.