TITLE: Does there exist a continuous function of compact support with Fourier transform outside L^1?
QUESTION [27 upvotes]: Let f be a complex-valued function of one real variable, continuous and compactly supported. Can it have a Fourier transform that is not Lebesgue integrable?

REPLY [2 votes]: Choose
$$f(x)=
\begin{cases}
\dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\
0,&\text{otherwise}
\end{cases}$$
Then $$\operatorname*{Im} \int_{0}^{\infty} dk \ e^{-\varepsilon k}
\int_{-\infty}^{\infty} dx \ f(x) e^{-ikx} = \int_{0}^{\frac12}
dx \ \frac{\frac12 -x}{\log(x)} \cdot
\frac{-x}{\varepsilon^{2}+x^{2}} \to \infty\text{ for }
\varepsilon \downarrow 0.$$
Therefore the Fourier transform of $f$ is not in $L^{1}$ .