Let $f\in L^{1}(\mathbb{R})$ that satisfies $$\int_0^\infty e^{itz}\widehat{f}(t)dt=0,\forall z\in \mathbb{H},$$ where $\mathbb{H}$ is the open upper half plane.
Does the above condition implies that $\widehat{f}$ vanishes identically on $[0,\infty)$?