Kahane-Katznelson-de Leeuw theorem and absolute convergence of Fourier series

We extend the Kahane-Katznelson-de Leeuw theorem to smoothness spaces by showing that for any $g\in W^{l,2}({\mathbb{T}}^d)$, there exists a function $f\in C^l({\mathbb{T}}^d)$ satisfying $\mathrm{\mid }\widehat{f}(n)\mathrm{\mid }\ge \mathrm{\mid }\widehat{g}(n)\mathrm{\mid }$ and $${\omega }_r(D^{l}f,t{)}_{\mathrm{\infty }}\approx {\omega }_r(D^{l}g,t{)}_2,t>0.$$ We apply this result to solve the Bernstein problem of finding necessary and sufficient conditions for the absolute convergence of multiple Fourier series. Finally, we explore the absolute integrability of Fourier transforms.