I have to compute the convolution of $ f(t) = \frac{1}{\pi}\frac{1}{t^2 + 1} $ with itself 4 times, i.e.
$$ f \star f \star f \star f $$
I slightly doubt that doing it in steps, i.e. taking $f \star f$ first and then taking the convolution of the result with f and then do the same once more, is the correct way of computing it..
Could anyone help me out with a hint of what method I should use?
Thank you!
Best Answer
You can use the fact that a convolution in the time domain is equivalent to a multiplication in the Fourier domain.
Thus if you carry out a Fourier transform of $f(t)$ as defined below:-
$$F(\omega)=FT(f(t))=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt$$ then $$F'(\omega)=FT(f(t)\ast f(t) )=F(\omega)F(\omega)$$ Therfore, to calculate the result of $f(t)\ast f(t)\ast f(t)\ast f(t)$ you would calculate the Fourier transform of $f(t)=\frac{1}{\pi}\frac{1}{1+t^2}$ similar to what is shown here for example, then multiply this result with itself three times, and finally apply the inverse Fourier transform.