I will preface this by saying I am certainly no expert in Probability theory.

My actual problem is an interpolation one, in which I am considering interpolation of bandlimited functions with shifts of some family of functions. One such family, with a given parameter $\alpha>0$ is the following

$$\phi_\alpha(x):=\dfrac{1}{(2\pi)^\frac{d}{2}}\int_{\mathbb{R}^d} e^{-\alpha\|\xi\|^p}e^{i\langle x,\xi\rangle}d\xi,\quad x\in\mathbb{R}^d.$$

In other words, $\phi_\alpha$ is the inverse Fourier transform of the function $e^{-\alpha\|\cdot\|^p}$.

Notation: $\|\cdot\|$ is the Euclidean distance on $\mathbb{R}^d$, and $\langle x,\xi\rangle$ is the usual dot product.

One can show that the interpolation scheme works (not important what that means at the moment) as long as $p>0$.

It was mentioned to me that this looks a lot like $p$-stable random variables in Probability theory. However, I noticed in looking up the definition, $p$-stable is usually only defined for $0<p\leq2$.

Is there a notion of this for $p>2$, or if not, why?

It may well be that there is no connection here, but it was interesting to me that all of the functions I considered could somewhat be related to probability distributions (e.g. Gaussians, and inverse multiquadrics of the form $$\psi_c(x) := \dfrac{1}{(\|x\|^2+c^2)^\beta},$$ where $\beta>d/2$. For $d=1$ and $\beta=1$, this is the Poisson kernel).

  • $\begingroup$ If I remember right, $\phi_\alpha$ is a nonnegative function iff $p\le 2$. $\endgroup$ – user147263 Nov 18 '14 at 1:16

No, thay cannot be naturally generalised to exponents $p>2$.

A random variable $X$ is $p$-stable if whenever we have a sequence $(X_n)_{n=1}^\infty$ of independent copies of $X$ then for any finite sequence of scalars $(\alpha_n)_{n=1}^N$ the random variables $$\sum_{n=1}^N \alpha_n X_n\quad\text{ and }\quad\Big(\sum_{n=1}^N |\alpha_n|^p\Big)^{1/p}X$$ have the same distribution. This implies that for any $r<p$ the sequence $(X_n)_{n=1}^\infty$ spans an isometric copy of $\ell_p$ in $L_r(\mathsf{P})$. However each closed subspace of $L_2(\mathsf{P})$ is a Hilbert space, so you cannot embed $\ell_p$ into $L_2(\mathsf{P})$ unless $p=2$.


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