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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 the symmetric stable lawlaws. 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
show/hide this revision's text 2 deleted 1 characters in body

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 the symmetric stable law. 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 and or references.

Anand
show/hide this revision's text 1

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 the symmetric stable law. 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 and references.

Anand