One could think at this point that the last definition may be extended to the
functions acting
for some
and
.
Even though such extension is formally possible it would be of little use
because the main tools of the following analysis are convolution and Fourier
transform.
Example
Consider the
given by the step
function


(step function)

The derivative
is called Dirac's delta
function:


(Dirac delta function)

We will use the
notation
The Fourier transform and the convolution extend to
similarly to the differentiation.
