Does anyone have a source or answer for whether Fourier inversion, in the sense that $$\mathcal{F}^{-1}(\mathcal{F}(f)) = f,$$ is valid for all $f \in L^p(\mathbb{R}^d)$?
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How do you define $\mathcal{F}(f)$ ? (look at $(f(x) e^{-\pi |x|^2/n^2}) \ast n^d e^{-\pi n^2 |x|^2}$) – reuns Oct 10 '17 at 05:28
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$\mathcal F^{-1}(\mathcal F(f))=f$ is true for all tempered distributions $f\in\mathcal S'(\mathbb R^d)$, including functions in any $L^p$ space when interpreted as tempered distributions. See for example Hunter's Applied Analysis, in the section on "Distributions and the Fourier Transform".
This goes outside the realm of $L^p$ spaces. One can define operators $\mathcal F$ and $\mathcal F^{-1}$ from $L^p$ to $L^q$ for $1\leq p\leq 2$ and $\frac 1 p + \frac 1 q=1$, agreeing with the transforms as tempered distributions. But for $p>2,$ the Fourier transform generally won't lie in $L^q$, and similarly for $p<2$ the inverse Fourier transform doesn't define an operator $L^q\to L^p.$ For example the Fourier transform of the $L^\infty$ constant $1$ function is not in $L^1.$
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