Let $\alpha$ be real numbers and let $f\colon\mathbb{R}\to \mathbb{C}$ be a function in $L^2 (\mathbb{R})$ (actually smooth and compactly supported, but this doesn't seem to be relevant). I am interested in the following integral average:
$\frac{1}{R}\int_{0}^{R}\int_{\mathbb{R}}\bar{f}\left(x\right)e^{2\pi i\alpha rx}f\left(x+r\right) dxdr$
and in its convergence as $R \to \infty$. I have reason to believe that it converges to zero, but offhand I don't see why it converges at all. Does anyone have any ideas about that?
Thanks for any help.