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We consider $f : [a, b] \subset \mathbb{R} \rightarrow D \subseteq \mathbb{C}$ and support may consist of only one point.

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All functions that vanish outside a set of measure zero are mapped to the zero function by the Fourier transform, and by the uniqueness theorem these are the only ones that are mapped to zero. – AD. Dec 25 '12 at 11:37
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Yes, it can, any function which has a null set as support has zero Fourier transform. This is easy to see from the fact that the Fourier transform is a unitary operator from $L^2$ to $L^2$.

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