nLab
derivative of a distribution

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 X nX \subset \mathbb{R}^n be an open subset of Euclidean space of dimension nn. For α n\alpha \in \mathbb{N}^n a multi-index, write

α α 1 α 1x 1 α n α nx n \partial_\alpha \coloneqq \frac{\partial^{\alpha_1}}{\partial^{\alpha_1} x^{1}} \cdots \frac{\partial^{\alpha_n}}{\partial^{\alpha_n} x^{n}}

for the corresponding operation of differentiation and write

|α|i=1nα i {\vert \alpha \vert} \;\coloneqq\; \underoverset{i = 1}{n}{\sum} \alpha_i

for the total degree.

For u𝒟(X)u \in \mathcal{D}'(X) then its derivative αu𝒟(X)\partial_\alpha u \in \mathcal{D}'(X) of order α\alpha is defined by

αu,b=(1) |α|u, αb \langle \partial_\alpha u, b \rangle \;=\; (-1)^{\vert \alpha \vert} \langle u, \partial_\alpha b\rangle

for any bump function bb, where αb\partial_\alpha b 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 u𝒟( n)u \in \mathcal{D}'(\mathbb{R}^n) a distribution and α n\alpha \in \mathbb{N}^n a multi-index with D αD^\alpha denoting the corresponding partial derivative, then

WF(D αu)WF(u). WF(D^\alpha u) \subset WF(u) \,.

(Hörmander 90, (8.1.10), p. 256)

Examples

Proposition

The distributional derivative of the Heaviside distribution Θ𝒟()\Theta \in \mathcal{D}'(\mathbb{R}) is the delta distribution δ𝒟()\delta \in \mathcal{D}'(\mathbb{R}):

Θ=δ. \partial \Theta = \delta \,.
Proof

For bC c ()b \in C^\infty_c(\mathbb{R}) any bump function we compute:

Θ(x)b(x)dx =Θ(x)b(x)dx = 0 b(x)dx =(b(x)| xb(0)) =b(0) =δ(x)b(x)dx. \begin{aligned} \int \partial\Theta(x) b(x) \, d x & = - \int \Theta(x) \partial b(x)\, dx \\ & = - \int_0^\infty \partial b(x) d x \\ & = - \left( b(x)\vert_{x \to \infty} - b(0) \right) \\ & = b(0) \\ & = \int \delta(x) b(x) \, dx \,. \end{aligned}
Example

Every point-supported distribution uu, hence with supp(u)={p}supp(u) = \{p\} for some point pp, is a finite sum of multiplies of derivatives of the delta distribution at that point:

u=α n|α|kc α αδ(p) u = \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum} c^\alpha \partial_\alpha \delta(p)

where {c α} α\{c^\alpha \in \mathbb{R}\}_\alpha, and for kk \in \mathbb{N} the order of uu.

(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 11:37:49. See the history of this page for a list of all contributions to it.