Fourier Inversion Theorem for Non-absolutely integrable functions(Ex: Sinc)

Question

I am confused on when you can apply the Fourier transform inversion theorem and what can be said about the inverse Fourier transform of a non-absolutely integrable function.

For example, the Fourier transform of the rectangle is the sinc function, the standard inversion theorem does not apply as sinc is not absolutely integrable.

So can the sinc function be inverted? Is the inverse unique? What are the facts regarding this and more generally, what are the facts regarding the inverse of Fourier transforms which are not absolutely integrable?

Can the Riemann integral be used to define them? How does it work(the proof of the inversion theorem relies heavily on DCT)?

My analysis texts do not address this problem, please direct me to a reference which does.

What is the L^2 inversion though? Again I haven't found any theorems that really address the specific issue of inversion. — maxical, Dec 24 '16 at 20:49
The $L^2$ inversion is essentially the same as the inversion for integrable Fourier transforms, only since the integral doesn't (usually) exist as a Lebesgue integral, you interpret it as a principal value integral, $\lim\limits_{R\to +\infty} \int_{-R}^R \dotsc$, with the limit taken in $L^2$ ($\int_{-R}^R\dotsc$ defines a function in $L^2 \cap C_0$). — Daniel Fischer, Dec 24 '16 at 20:57
Do you have a proof showing that the improper integral defines a function? — maxical, Dec 25 '16 at 03:13
Possible duplicate of Inverse Fourier transform relation for $L^2$ function — , Dec 30 '16 at 03:54
Also, internet search for "fourier inversion L^2" brings up a bunch of sources such as this one — , Dec 30 '16 at 03:54

Fourier Inversion Theorem for Non-absolutely integrable functions(Ex: Sinc)

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