Applied linear operators and spectral methods/Differentiating distributions

Differentiation of a distributionEdit

If   is a differentiable function in   whose first derivative is locally integrable, then the derivative   defines its own distribution

 

Integrating by parts and noting that at the boundaries  ,   because of compact support, we have

 

This suggests defining the derivative of any distribution   via

 

We need to check that this definition satisfies the properties of a distribution. We observe that if   is a test function then so is  . Linearity is obvious. What about continuity? If   is a zero sequence, so is the sequence   and thus   tends to zero as  . Hence (1) defines a distribution.

Remark: All distributions are infinitely differentiable is the class of test functions includes only infinitely differentiable functions.

Comment: One can take test functions which are only   differentiable. Then the distribution will only be   times differentiable. Thus, enlarging the class of test functions reduces the class of distributions.

More generally, if   is the  th derivative of the distribution  ,

 

Distributions can be generated by functions which are not differentiable in the ordinary sense. However, we can differentiate them in the distribution sense.

For example, the derivative of the delta distribution gives us the dipole distribution:

 

Also, the derivative of the Heaviside function is given by

 

Since  , from the fundamental theorem of calculus,

 

Therefore the derivative of the Heaviside function is the delta function. Template:Lectures