Absolute Fourier convergence from a square-integrable derivative (${\sum }_{n\ne 0}|\hat{f}(n)|\le (\pi /\sqrt{3})\Vert f^{\prime }{\Vert }_2$)

For a periodic absolutely continuous function with derivative in $L^2$, integration by parts gives $\widehat{f^{\prime }}(n)=in\hat{f}(n)$. 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.