(a) Give the definition of the Fourier transform $\widehat{f}$ of a function $f \in L^{1}\left(\mathbb{R}^{d}\right)$.

(b) Explain what it means for Fourier inversion to hold.

(c) Prove that Fourier inversion holds for $g_{t}(x)=(2 \pi t)^{-d / 2} e^{-\|x\|^{2} /(2 t)}$. Show all of the steps in your computation. Deduce that Fourier inversion holds for Gaussian convolutions, i.e. any function of the form $f * g_{t}$ where $t>0$ and $f \in L^{1}\left(\mathbb{R}^{d}\right)$.

(d) Prove that any function $f$ for which Fourier inversion holds has a bounded, continuous version. In other words, there exists $g$ bounded and continuous such that $f(x)=g(x)$ for a.e. $x \in \mathbb{R}^{d}$.

(e) Does Fourier inversion hold for $f=\mathbf{1}_{[0,1]}$ ?