§ Derivative of step is dirac delta
I learnt of the "distributional derivative" today from my friend, Mahathi.
Recording this here.
§ The theory of distributions
As far as I understand, in the theory of distributions, A distribution
is simply a linear functional .
The two distributions we will focus on today are:
- The step distribution,
- The dirac delta distribution, .
- Notationally, we write where is a distribution, is a function as .
- We can regard any function as a distribution, by sending to .
- But it also lets us cook up "functions" like the dirac delta which cannot actually exist as a function. So we move to the wider world of distributions
§ Derivative of a distribution
Recall that notationally, we wrote as . We now want
a good definition of the derivative of the distribution, . How? well,
use chain rule!
Here, we assume that , ahd that is differentiable at .
This is because we only allow ourselves to feed into these distribtions
certain classes of functions (test functions), which are "nice". The test
functions (a) decay at infinity, and (b) are smooth.
The derivation is:
- Thus, the derivative of the step distribution is the dirac delta distribution.