The prototypical example of a distribution is the Dirac delta function, defined as a linear functional taking a well behaved test function $\phi:\mathbb{R} \to \mathbb{R}$ and returning its value at the origin, $$(\delta,\phi) = \phi(0).$$ Intuitively, one can imagine $\delta$ as a sharply peaked function around the origin so that $$(\delta, \phi) = \int_{\mathbb{R}} \delta(x) \phi(x) dx.$$ A lot of properties of the delta functions are motivated by considering the above integral and performing various partial integrations etc. Then one uses the property that the test functions, together with all of their derivatives, vanish at the boundary of integration, which is infinity.
However, in physical applications, one often has to consider distributions for which the test functions are defined on a different domain $D$. For example:
- $D = [0, \infty)$: for spherical symmetric problems, physical quantities depend only on the radial variable $r$ which is inherently non-negative. Therefore, the test functions of the radial variable have positive reals as their domain, and the radial delta function $\delta_r$ should(?) exist with the fundamental property $$(\delta_r, \phi) = \int_0^\infty \delta_r (r) \phi(r) dr = \phi(0).$$ However, due to the non-trivial boundaries, the properties of the radial delta function should be different from the usual delta function. E.g. we cannot perform the partial integrations so easily. The similar problem persists in other domains, such as
- $D = [0,1]$: as in the case of zenith angle variable in spherical coordinates;
- $D = [0,2\pi)$: with $\lim_{x \to 2\pi} \phi(x) = \phi(0)$; the periodic boundary conditions, as in the case of azimuth angle variable in spherical coordinates.
How does one approach distributions on these domains? Can they be simply related to the usual distributions on unbounded domains?