On the constant in a transference inequality for the vector-valued Fourier transform

TL;DR

This paper explicitly computes the constant involved in a transference inequality for the vector-valued Fourier transform, clarifying a previously implicit aspect of the proof of Fourier type equivalence.

Contribution

It provides an explicit calculation of the constant in the transference inequality, which was previously only described implicitly.

Findings

Explicit value of the constant in the transference inequality

Clarification of the proof of Fourier type equivalence

Enhanced understanding of the sinc function sum involved

Abstract

The standard proof of the equivalence of Fourier type on R^d and on the torus T^d is usually stated in terms of an implicit constant, defined as the minimum of a sum of powers of sinc functions. In this note we compute this minimum explicitly.