If $f \geq 0$, $f=0$ outside $(-1/2,1/2)$, and $f \geq cx^{1/2}$ on $(-1/4,1/4)$, is it possible to have $|\hat{f}(k)| = O(|k|^{-1})$? How fast could $\hat{f}(k)$ goes to zero?
2026-05-14 07:37:25.1778744245
Function bounded below by square root near 0. How fast can Fourier transform go to zero?
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