Van der Corput inequality for finite scalar sequences ($|N^{-1}{\sum }_n{u}_n{|}^2$)

For complex numbers $|u_n|\le 1$, $0\le n<N$, and $1\le H\le N$, put $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. ThenExtend 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.