If I understood well Riemann-Lesbegue lemma it says that the Fourier transform $\hat{f}(\xi)$ of a function $f(x)$, $x\in\mathbb{R}$ decays to zero with $\xi\to\infty$. Furthermore the more $f$ has continuous derivative the faster its Fourier transform goes to zero. In particular a discontinuous e.g. $f(x)=\text{Heaviside}(|a^2-x^2|)$ will go as $|\hat{f}(\xi)|\propto \xi^{-1}$. A continuous but not continuous first derivative function $f(x)=e^{-|x|}$ will go as $|\hat{f}(\xi)|\propto \xi^{-2}$. For $C^\infty(\mathbb{R})$ function the decays is faster than any power law.

Now I believe that Paley-Wiener theorem says that for analytic function $f$, $|\hat{f}(\xi)|= O( e^{-\alpha \xi})$ for some $\alpha$. (Here I am not sure it is valid for $\mathbb{R}$).

My question is how to distinguish smoothness/regularity of two analytic function on $\mathbb{R}$? For example $g(x)=e^{-x^2/(2\sigma^2)}$ and $h(x)=1/(1+(x/\sigma)^2)$. From Fourier transform we know that $\hat{g}\propto e^{-(\sigma\xi)^2/2}$ and $\hat{h}\propto e^{-\sigma\xi}$.

So do it mean that Gaussian function $g$ is "more than analytic" than Cauchy $h$? How to understand this very fast decay in terms of Paley-Wiener theorem or Riemann-Lesbegue lemma? Is there other means to measure regularity of functions than Fourier transform? I have read about modulus of continuity $w$ (see one example of definition), but so far I have not manage to compute it (analytically or numerically) for $g$ or $h$ (numerically I always find $w(1/\xi)\propto 1/\xi$ which does not reflect the Fourier transform).


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