We consider $f : [a, b] \subset \mathbb{R} \rightarrow D \subseteq \mathbb{C}$ and support may consist of only one point.
Can fourier transform of a function with not empty support be zero on whole range?
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complex-analysis
fourier-analysis
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0All 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. – 2012-12-25
1 Answers
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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 $L^2$.