Existence of a non-unique Inverse Fourier transform

Question

Can someone give an example of an inverse Fourier transform that's not unique and show it? Thank you for any help.

Answer 1

The big question is : how do you define the Fourier transform and the inverse Fourier transform ? I ask this because you can define several set of functions (or class of equivalence of functions) where the Fourier and inverse Fourier transform is well-defined. Outside of those sets, the Fourier transform is not so well and uniquely defined.

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In general you start from $\hat{f}(\xi) = \int_{-\infty}^\infty f(x)e^{-2i \pi \xi x}dx$ well-defined whenever $f \in L^1$. You prove the convolution theorems and the Fourier inversion theorem $f(x) = \int_{-\infty}^\infty \hat{f}(\xi)e^{-2i \pi \xi x}d\xi$ for $f,\hat{f} \in S(\mathbb{R})$ (the Schwartz space in particular for $f,\hat{f}$ Gaussian) and extend it to $f,\hat{f} \in L^1$.

Then, using that $L^1\cap L^2$ is dense in $L^2$, you can extend all this to $f \in L^2$ (a Hilbert space where the Fourier transform is a unitary operator, see the Parseval's theorem).

Finally from $S(\mathbb{R})$ and the Fourier transform on $S(\mathbb{R})$ you can define the tempered distributions $S'(\mathbb{R})$ (containing the Dirac delta $\delta$ and its derivatives, useful for solving ordinary differential equations) and the Fourier transform and the Fourier inverison theorem on $S'(\mathbb{R})$.