A characterization of Banach spaces having a nontrivial Fourier type

Abstract

Let $X$ be a Banach space. We show that $X$ has a nontrivial Fourier type if and only if there exists (equivalently, for all) $~0~<~\alpha~<~1$, such that for all $f\in~C^\alpha([0,~2\pi];~X)$ satisfying $f(0)~=~f(2\pi)$, we have $\sum_{n\in{\mathbb~Z}}\Vert~\hat~f(n)\Vert^{1/\alpha}<~\infty$.