Assume you have a real-valued function $f(x)$ defined over whole $\mathbb{R}$ and $f \in L^2(\mathbb{R})\cap L^1(\mathbb{R})$.

What additional characteristics should this function have in order that its Fourier transform $\hat{f}(k)$ decays exponentially at infinity? Should $f$ be smooth or analytic or both?

With exponential decay I mean that $\hat{f} \sim e^{-Ck}$ up to some multiplicative factor of the form $k^p$ and for some positive constant $C$. In other words, that the behavior of $\hat{f}$ for $k\rightarrow\infty$ is essentially given by $\exp(-Ck^\alpha)$ with $\alpha = 1 + \epsilon$ and $|\epsilon|$ arbitrarily small. I have read some similar questions here and their answers but this made me even more confused.

The problem is also the counter example that the Gauss function $e^{-x^2}$ belongs to $L^2(\mathbb{R})\cap L^1(\mathbb{R})$, it is both smooth and analytic but its Fourier transform is also a Gauss function, i.e., it decay faster that exponentially.


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