I'm working through some problems of Fourier Analysis by Stein, Shakarchi and I got stuck trying to solve the following problem:
Let $S_N = \sum_{n=1}^N e^{2\pi i f(n)}$. Show that for $H\le N$, one has
$|S_N|^2 \le c \frac NH \sum_{h=0}^H\, \left|\,\sum_{n=1}^{N-h} e^{2\pi i(f(n+h)- f(n))} \right|$ for some constant $c>0$ independent of $N$, $H$, and $f$.
There is even a hint: Let $a_n = e^{2\pi if(n)}$ if $1\le n\le N$ and $0$ otherwise. Then write $H\;\sum_n a_n = \sum_{h=1}^H \sum_n a_{n+h}$ and apply the Cauchy-Schwarz inequality.
Well, I'm realy horrible at this stuff, but I'll write down at least a beginning
$\begin{align} |S_N|^2 &= \frac 1{H^2} \left| \sum_{h=1}^H \sum_n a_n \right|^2 \\ &\leq \frac1H \sum_{h=1}^H \left|\sum_n a_{n+h} \right|^2 \\ &= \frac1H \sum_{h=1}^H \sum_{n,m} a_{n+h}\overline{a_{m+h}} \end{align}$
and after this I already get lost. =( (Of course, I have scribbed down much more than just this, but it all lead nowhere, so I won't expose you to that...)
I would really appreciate some help. Cheers!