Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is an integrable function with Fourier coefficients given by $\hat{f}$. Then, since $|f|^2 = f \cdot \bar{f}$, we have:
$$\displaystyle \int_{0}^{2\pi} |f(k)|^2 \mathrm{d}k = \int_{0}^{2\pi}\left( \sum_{a=0}^{\infty} \hat{f}(a)e^{iak} \right)\left( \sum_{b=0}^{\infty} \hat{f}(b)e^{-ibk} \right)\mathrm{d}k.$$
Using the Cauchy product, we obtain:
$$\displaystyle \int_{0}^{2\pi} \sum_{c=0}^{\infty}\sum_{l=0}^{c}\hat{f}(l)\hat{f}(c-l)e^{i(l - c + l)k}\mathrm{d}k.$$
Now, the integral is zero unless $2l = c,$ in which case, integrating the exponential term (the only part of the series which depends on $k$), gives us $2\pi.$ Imposing this condition leads to the sum:
$$\displaystyle 2\pi \sum_{c=0}^{\infty}\sum_{l=0}^{c}(\hat{f}(l))^2.$$
My question is this: can we do anything more here, using the condition that $2l = c$?