$C_b^0(\mathbb{R})=\{f\in C(\mathbb{R}),\lim\limits_{|x|\to\infty}f(x)=0\}$
So can we construct an function in $C_b^0(\mathbb{R})$ such that it's Fourier transform is not in $L^1(\mathbb{R})$?
What kinds of sufficient property does it have?
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I would try something like $|x|e^{-x^2}$ or other related functions. I'm not sure that this will work though. Also, see this mathoverflow thread: http://mathoverflow.net/questions/3764/does-there-exist-a-continuous-function-of-compact-support-with-fourier-transform – Moya Dec 05 '15 at 15:25
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@Moya Not possible because $|x|e^{-x^2}$ is moderate decrease. In fact such example should not be $L^2$. – Liding Yao Dec 08 '15 at 14:54