I'd like to prove that Fourier coefficient of two continuous and periodic functions, with a period of $1$, the following equality holds:
$\widehat {f*g}(n)=\hat f(n)\cdot \hat g(n)$
( $\hat x(n)$- Fourier coefficient of $x$, and $y*z$- Convolution of $y$ and $z$)
If I knew that $f, g$ are continually differentiable I could have used the fact that Fourier series $f,g$ converge to the actual functions, and I could use it to prove the claim. How can I prove it with these conditions and without using the order of the integrals, cause I am able to prove with that, but I am not allowed to.
Thanks a lot