= Squared derivative correlations force frequency-graph energy
{title2=$\sum_{k\in B}|\widehat{\Delta_kf}(\phi(k))|^2\ge\alpha N^3\Longrightarrow E(\operatorname{graph}\phi)\ge\alpha^4N^3$}
Use the unnormalized <Fourier transform> and $|f|\le1$. Normalize each derivative correlation by $N$ and align its phase. Their absolute sum is at least $\alpha N$. One <Cauchy-Schwarz inequality>, expansion by shift/frequency differences, and a second <Cauchy-Schwarz inequality> bound its fourth power by $N$ times the <additive energy of a frequency graph>. <Parseval identity on a finite group> supplies the factor $N$; exact equality of both shift sums and frequency sums is retained.
Back to article page
