We know that if the integrable function $f\in H^\alpha(\mathbb{R}), 0<\alpha<1$ (Hölder continuous), then its Fourier transform $\hat{f}$ has the asymptotic form $ O (1/x^\alpha)$ as $x\to\infty$. We also know that if an integrable function $f\in BV(\mathbb{R})$ (has bounded variation on $\mathbb{R}$), then its Fourier transform $\hat{f}$ has the asymptotics $O (1/x)$ as $x\to\infty$. If $f(x)=\chi_{[-1,1]}(x)$, then its Fourier transform has asymptotic behavior $O(1/x)$ as $x\to\infty$, and we can conclude that the first statement is only a necessary condition, but not a sufficient one. Are there any least conditions on the rapid decay of $\hat{f}$ that guarantee that $f$ belongs to the class $H^\alpha$ or $BV$?
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