Consider a function $f$ which is continuous on $\mathbb{R^n}$ with compact support. Does its Fourier transform $\hat{f}(\xi) = \int_{\mathbb{R^n}} f(x)e^{-2 \pi i x \cdot \xi} \,dx$ always lie in $\mathbb{L}^{1}(\mathbb{R^n})$ ?
I would say yes if f is $k^{th}$ differentiable for sufficiently large k. But what if f is merely contiuous? I think in this case its Fourier transform can decay slowly, but I fail to find a counterexample.
Any hint is highly aprreciated!!!