Support of a convolution with the help of Titchmarsh theorem

Question

I have to use Titchmarsh theorem in order to prove that : if $f\in L^1[-1,1]$, and $supp(f*f*f*f-f*f)\subset [-1,1]$ then $supp(f)\subset[-1/4,1/4]$. Does anyone have an idea ? Thank you.

Answer 1

If $supp\, f\subset [a,b]$ and $supp\,g\subset [c,d]$ then $supp\, (f*g)\subset [a+c,b+d]$ and thus $supp\,(f*f)\subset [2a,2b]$ and $supp (f*f*f*f)\subset [4a,4b]$ and thus $$ supp\,(f*f*f*f-f*f)\subset [\min(2a,4a), \max(2b, 4b) ]. $$