I am trying to compute the following quantity:
$$ \sum_{k=-\infty}^{\infty} 2^{k \frac{1}{2}}\left\{\int_{\Gamma_k}|\widehat{f}(x)|^2 d x\right\}^{\frac{1}{2}}$$
where: $ \Gamma_k=\left\{x \in \mathbb{R}: 2^{k-1}<\left|x\right|<2^k\right\}$ and:
$$ f(x) = \delta^{-1} x^{1/4} 1_{(\delta,M)}(x)$$
where $\delta$ and $M$ are costants. Any idea on how to tackle this problem?