Fourier transform of $\chi_B$ not in $L^1(\mathbb R^n)$ for the unit ball $B$

Question

Let $B$ be the unit ball in $\mathbb R^n$. Show that $\hat{\chi_B}\not\in L^1(\mathbb R^n)$. Is it possible to show this without computing the Fourier transform explicitly? This means I want to show that $\|\hat{\chi_B}\|_1=\int_{\mathbb R^n}|\int_{\mathbb R^n}\chi_B(x)e^{-2\pi i\langle x,\xi\rangle}dx|d\xi\geq \infty$

Answer 1

If $f$ and $\hat f$ are both integrable then $f$ is almost everywhere equal to a continuous function. [This is a consequence of the Inversion Formula]. In this case $\chi_B $ is not almost everywhere equal to a continuous function. Since $\chi_B$ is integrable it follows that $\hat {\chi_B}$ cannot be integrable.