The paper "On the constant in a transference inequality for the vector-valued Fourier transform" revisited

The standard proof of the equivalence of Fourier type on ${\mathbb{R}}^d$ and on the torus ${\mathbb{T}}^d$ is usually stated in terms of an implicit constant which can be expressed in terms of the global minimiser of the functions $$f_r(x)=\sum _{m\in \mathbb{Z}}{\mid \frac{\sin (\pi (x+m))}{\pi (x+m)}\mid }^{2r},x\in [0,1],r\ge 1.$$ The aim of this note is to provide a short proof of a result of the authors which states that each $f_r$ takes a global minimum at the point $x=\frac{1}{2}$.