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tensor product of distributions
Contents
Context
Functional analysis
Overview diagrams
Basic concepts
Theorems
Topics in Functional Analysis
Contents
Idea
The tensor product of distributions is the generalization to distributions of the tensor product of smooth functions, hence it defines for two distributions and a new distribution on the Cartesian product space which, as a generalized function behaves like .
Definition
Definition
(tensor product of smooth functions)
For two open subsets of some Cartesian space, there is an injection from the tensor product of the real vector spaces of smooth functions on the separate spaces to that on the Cartesian product space:
with
Proposition
(tensor product of distributions)
Let and be distributions. Then there is a unique distribution of two variables such that for all pairs of bump functions and its value on their tensor product according to def. is
This is called the tensor product of with
(HΓΆrmander 90, theorem 5.1.1)
Properties
Example
(wave front set of tensor product distribution)
Let and be two distributions. then the wave front set of their tensor product distribution (def. ) satisfies
where denotes the support of a distribution.
(HΓΆrmander 90, theorem 8.2.9)
References
- Lars HΓΆrmander, section 5.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990 (pdf)
Last revised on October 24, 2017 at 12:57:54.
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