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

Abstract: 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|^{2r},\qquad 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 = \frac12).