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derivative of a distribution
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Contents
Context
Functional analysis
Overview diagrams
Basic concepts
Theorems
Topics in Functional Analysis
Contents
Idea
The concept of derivative of a distribution is the generalization of the concept of derivative of a smooth function with distributions thought of as generalized functions. The concept is uniquely fixed by enforcing the formula for integration by parts to extend from integrals against compactly supported densities to distributions.
Definition
Let be an open subset of Euclidean space of dimension . For a multi-index, write
for the corresponding operation of differentiation and write
for the total degree.
For then its derivative of order is defined by
for any bump function , where is the ordinary derivative of smooth functions.
(e.g. Hörmander 90, def. 3.1.1 with (3.1.1)’)
Properties
Proposition
(derivative of distributions retains or shrinks wave front set)
Taking derivatives of distributions retains or shrinks the wave front set:
For a distribution and a multi-index with denoting the corresponding partial derivative, then
(Hörmander 90, (8.1.10), p. 256)
Examples
Proposition
The distributional derivative of the Heaviside distribution is the delta distribution :
Proof
For any bump function we compute:
Example
Every point-supported distribution , hence with for some point , is a finite sum of multiplies of derivatives of the delta distribution at that point:
where , and for the order of .
(e.g. Hörmander 90, theorem 2.3.4)
References
- Lars Hörmander, The analysis of linear partial differential operators, vol. I, Springer 1983, 1990
Last revised on November 7, 2017 at 16:37:49.
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