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Let $\phi\in C_{0}^{\infty}(\mathbb{R}^n)$ and equals to 1 near the origin,then show that $e^{i|\xi|}\phi(\frac{|\xi|}{\mu})$ is a fourier transform of an $L^1$ function with any $\mu>0$,and how its $L^1$-norm depend on $\mu$(does its norm goes $\infty$,when $\mu\to\infty$ ? and how fast ?) .

If $f\in H^k$,and $k>\frac{n}{2}$,then it belongs to $\mathcal{F}L^1$.My question is besides this,are there other useful methods to show that a given function (especially when it's not differentiable at some points)is a fourier transform of an $L^1$ function ?

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