Distribution - Imported Knowlpedia Content

Definition

Let $\Omega\subseteq\mathbb R^n$ be open and let $\mathcal D(\Omega)=C_c^\infty(\Omega)$ carry its canonical locally convex topology. A distribution on $\Omega$ is a continuous linear functional

T:\mathcal D(\Omega)\to\mathbb C.

Thus the space of distributions, denoted $\mathcal D'(\Omega)$ , is the topological dual of the test-function space. Continuity means that on functions supported in each fixed compact $K\subset\Omega$ , $T$ is bounded by finitely many uniform derivative seminorms. This continuity condition distinguishes distributions from arbitrary algebraic linear functionals.

Regular and singular examples

Every $f\in L^1_{\mathrm{loc}}(\Omega)$ defines a regular distribution by

T_f(\varphi)=\int_\Omega f(x)\varphi(x)\,dx.

The Dirac distribution $\delta_a(\varphi)=\varphi(a)$ for $a\in\Omega$ is singular: it cannot be represented by a locally integrable function. Both satisfy the same continuity requirement.

Differentiation and support

For a multi-index $\alpha$ , the distributional derivative is defined by

\langle \partial^\alpha T,\varphi\rangle=(-1)^{|\alpha|}\langle T,\partial^\alpha\varphi\rangle.

This integration-by-parts definition makes every distribution infinitely differentiable in the distributional sense. The support of $T$ is the complement of the largest open set on which $T$ vanishes on all test functions.

Conventions and scope

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd edition, Springer, 2003. DOI record. Relevant: chapters 1–3 on test functions, distributions, and differentiation.
François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967; Dover reprint, 2006. Publisher record. Relevant: chapters on test-function spaces and distributions.