Linear Independence of Time-Frequency Translates in \mathbb {R}^d

We establish the linear independence of time-frequency translates for functions \(f\) on \(\mathbb {R}^d\) having one-sided decay \(\lim _{x \in H,\ |x|\rightarrow \infty } |f(x)| e^{c|x| \log |x|} = 0\) for all \(c>0\), which do not vanish on an affine half-space \(H \subset \mathbb {R}^d\).