I have noticed that this result is hinted at in several books, but am having trouble proving it:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f, \hat{f} \in C_c^\infty(R^n) \Rightarrow f \equiv 0. [/tex]

in other words, if both f and its fourier transform

are smooth, compactly supported functions on n-dimensional euclidean space

then f is identically zero.

any advice? i thought of using the fourier inversion theorem, which tells me that f agrees almost everywhere (Lebesgue) with the continuous function

[tex]f_0 = (\hat{f})\check{} = (\check{f})\hat{}[/tex]

and then showing that one (or both) of those are zero; continuity of f would then take care of the "almost everywhere" part. but i'm not really sure what to do.

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# Fourier transform of f \in C_c^\infty(R^n)

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