$f\in L^{\infty}(\mathbb{R})$ as Fourier transform

Question

I need to know if one can view a function $f\in L^{\infty}(\mathbb{R})$ as a Fourier transform of a certain function, say g?

If the answer is positive please state the proof, or help me find one. Thanks

Answer 1

In general, no. For example, take $f(\xi) = 1$. There is a generalization of the Fourier transform, the Fourier–Stieltjes transform, and that maps a Dirac delta to $f(\xi) = 1$. So, in general, this inverse could be a distribution.

$L^1(\mathbb{R})$ is pretty much the largest class of functions which we can define the Fourier transform for. The Fourier transform of any function in $L^1(\mathbb{R})$ must go to zero at $\pm \infty$.

Edit to make that last point clearer: any function which does not go to zero at infinity will not be the Fourier transform of some function.

Another edit: See the comments; saying "$L^1(\mathbb{R})$ is the largest class..." was careless.