Denote by $C_{0}\left(\mathbb{R}^{n}\right)$ the space of continuous complex-valued functions on $\mathbb{R}^{n}$ converging to zero at infinity. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$.

(i) Prove that the image of $L^{1}\left(\mathbb{R}^{n}\right)$ under $\mathcal{F}$ is included and dense in $C_{0}\left(\mathbb{R}^{n}\right)$, and that $\mathcal{F}: L^{1}\left(\mathbb{R}^{n}\right) \rightarrow C_{0}\left(\mathbb{R}^{n}\right)$ is injective. [Fourier inversion can be used without proof when properly stated.]

(ii) Calculate the Fourier transform of $\chi_{[a, b]}$, the characteristic function of $[a, b] \subset \mathbb{R}$.

(iii) Prove that $g_{n}:=\chi_{[-n, n]} * \chi_{[-1,1]}$ belongs to $C_{0}(\mathbb{R})$ and is the Fourier transform of a function $h_{n} \in L^{1}(\mathbb{R})$, which you should determine.

(iv) Using the functions $h_{n}, g_{n}$ and the open mapping theorem, deduce that the Fourier transform is not surjective from $L^{1}(\mathbb{R})$ to $C_{0}(\mathbb{R})$.