I know the Fourier transform of a compactly supported function cannot be compacted supported. But what if the function is supported on $(0,+\infty)$ or $(-\infty,0)$? Is there any chance for its Fourier transform be compactly supported?
2026-04-03 06:10:56.1775196656
Could the Fourier transform of a one-sided non-negative function be compactly supported?
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The inverse Fourier transform of an $L^1$ function of compact support is an entire function. Therefore it can't be zero on any set with a finite limit point, in particular $(0,\infty)$ or $(-\infty,0)$.