Finding $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$

Question

I heard there were functions $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$. Is there a concrete example of such functions ? Thanks in advance !

Answer 1

Let $r > 0$. Then $$ \int_{0}^{\infty}e^{-rx}e^{-isx}dx=\frac{1}{r+is}. $$ The function $f(x)=e^{-rx}\chi_{[0,\infty)}(x)$ is in $L^1$, but $\hat{f}(s)=\frac{1}{\sqrt{2\pi}(r+is)}$ is not in $L^1$.