Convolution of two identical functions

Question

The general convolution of a function over the real line is defined as

$$(f\star g):=\int_{-\infty}^{\infty}f(t-h)g(h)dh$$

Can I say anything general about

$$f\star f?$$

Seems that there are no identities about that(or even properties) in standard literature.

Answer 1

Well, in terms of the Fourier transform (FT) $$\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$$ which is $$f*f =\mathcal{F}^{-1}\{ \mathcal{F}\{f\} ^2 \}$$ You could compute the FT of $f$, square that FT, then do the inverse FT.