Let us consider a $2 \pi-$periodic function $U(x)$. From Parseval's Theorem, I have the relation $$ \sum_{k} |U_{k}|^{2} = \int_{- \pi}^{\pi} \!\! \frac{\mathrm{d} x}{2 \pi} \, |U (x)|^{2} \, , $$ where the Fourier coefficients are defined as $$ U_{k} = \int_{\pi}^{\pi} \!\! \frac{\mathrm{d} x}{2 \pi} \, \mathrm{e}^{- \mathrm{i} k x} U (x) \, . $$ One virtue of Parseval's Theorem is that it allows me to compute the sum $\sum_{k} |U_{k}|^{2}$ by only computing one integral, without having to determine all the individual Fourier coefficients.
Now, I want to estimate a slightly different sum, given by $$ \sum_{k} |k| |U_{k}|^{2} $$ It is possible to write down a variant of Parseval's Theorem, so as to rewrite in a simple manner this sum?