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}}\left|\frac{\sin(π(x+m))}{π(x+m)}\right|