Consider the signal . What is the convolution of with itself?
From the definition of convolution , we have
Unfortunately, the latter integral is plagued by convergence problems. To make matters worse, is not even defined at . Fortunately, the former integral is a familiar one: it’s a scaled Hilbert transform of . It is well-known  that the Fourier transform of (the impulse response of a Hilbert transformer) is the following
where is the signum function. Thus, we have that the Fourier transform of is
Convolution in the time domain corresponds to multiplication in the frequency domain. Hence, the Fourier transform of the convolution is
and, taking the inverse Fourier transform, we finally obtain
where is the Dirac delta. This is a prime example of a problem that Signal Processing professors in engineering departments tend to love, but that drives (some) mathematicians up the wall.
 Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, Signals & Systems, 2nd edition, Prentice-Hall, 1996.