Asking for a Fourier inverse transform, which is related to stable laws Dear friends,
Denote the function 
$$
G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;.
$$
It is well known that if $a\in ]0,2]$, $G_a(x)$ is the density of symmetric stable laws. The problem is what happens if $a>2$ ? It is not non-negative any more. But $\int_R G_a(x)d x=1$. Where can I find more properties of this function? For example, how does it decay (modulo the small oscillation)?
Thank you very much for any hints or references.
Anand
 A: Precise estimates of the function $G_\alpha$ (also in the multi-dimensional case), its derivatives, and more complicated functions appearing as fundamental solutions of parabolic pseudo-differential equations with homogeneous symbols, can be found in Chapter 4 of the book:
S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Birkhauser, Basel, 2004.
A: See the article "Some theorems on stable processes" in
Trans. Amer. Math. Soc. vol. 95 (1960), pp. 263–273, by 
Blumenthal and Getoor. Apparently the computation of this Fourier transform
is due to Polya in "On the zeros of an integral function represented by Fourier's integral", Messenger of Math. vol. 52 (1923), pp. 185-188.   
A: I think you may want to look at this MSE question. Amusingly enough, this question was to prepare a lecture for when the asker was covering my class last fall. This kind of argument is mentioned in Durrett's Probability: Theory and Examples in Chapter 3.
