= Absolute Fourier convergence from a square-integrable derivative
{title2=$\sum_{n\ne0}|\widehat f(n)|\leq(\pi/\sqrt3)\|f'\|_2$}
For a periodic absolutely continuous <function> with derivative in $L^2$, <integration by parts> gives $\widehat{f'}(n)=in\widehat f(n)$. <Bessel's inequality> and the <Cauchy-Schwarz inequality> prove the displayed bound in the normalized circle norm. Adding the constant coefficient gives an absolutely and uniformly convergent <Fourier series>. Merely bounding individual coefficients by $O(1/|n|)$ would not establish absolute convergence.
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