= Van der Corput inequality for finite scalar sequences
{c}
{title2=$|N^{-1}\sum_n u_n|^2$}
For complex numbers $|u_n|\leq1$, $0\leq n<N$, and $1\leq H\leq N$, put $C_h=\sum_{n=0}^{N-h-1}u_{n+h}\overline{u_n}$. Then
$$
\left|\frac1N\sum_{n=0}^{N-1}u_n\right|^2
\leq\frac{N+H-1}{NH}\left(1+2\sum_{h=1}^{H-1}\left(1-\frac hH\right)\frac{|C_h|}{N}\right).
$$
Extend the sequence by zero, count each summand in $H$ consecutive windows, and apply the <Cauchy-Schwarz inequality> to their sum. Expansion of the squared window sums gives the stated coefficients. Fixing $H$ and making each normalized correlation tend to zero bounds the limiting squared average by $1/H$; then let $H$ tend to infinity. This is the scalar finite form underlying the <Van der Corput lemma>.
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