If $f,g \in L^2([-\pi,\pi])$ and are periodic with period $2\pi$ then the convolution is integrable. Can somebody comment whether my proof is correct?
$\int_{[-\pi,\pi]} |\int_{[-\pi,\pi]} f(y)g(x-y)dy|dx \leq \int_{[-\pi,\pi]} (\int_{[-\pi,\pi]} |f(y)||g(x-y)|dy)dx$ by triangle inequality. Now we focus on the inner integral (can iterate by tonnelli since non negative). $\int_{[-\pi,\pi]} |f(y)||g(x-y)|dy \leq \|f\|_{L^2{[-\pi,\pi]}} \sqrt{\int_{[-\pi,\pi]}|g(x-y)|^2 dy}.$ Now make a substitution $u= x-y$, then $du=-dy$ to get that the integral on the right equals $\sqrt{\int_{[x-\pi,x+\pi]}|g(u)|^2 du}$ But g periodic of period $2\pi$ so this simply equals $\sqrt{\int_{[-\pi,\pi]}|g(u)|^2 du}= \|g\|_{L^2([-\pi,\pi])}.$ So all together we get the double integral is less or equal to $2\pi \|f\|_{L^2([-\pi,\pi])} \|g\|_{L^2([-\pi,\pi])} <\infty$
Is this correct? I am hesitant as I know it is really trivial that the periodicity means the two integrals are equal but I haven't been able to show it mathematically. Otherwise is everything correct? Thanks.