The question has probabilistic origins, but it would take too long to  elaborate. $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\eS}{\mathscr{S}}$ 

Fix a nonnegative   Schwartz  function  $w:\bR\to \bR$. For any positive integer $m$ let $V_m:\bR^m\to\bR$  denote the Fourier transform of 

$$ w_m:\bR^m\to\bR,\;\;  w_m(x)=w(|x|^2/2). $$

More precisely

$$ V_m(\xi)=\int_{\bR^m} e^{-\ii (x,\xi)} w(|x|^2/2) dx. $$


$V_m$ is an $O(m)$-invariant Schwartz function on $\bR^m$ so that it  has the form  

$$ V_m(\xi)= f_m\Bigl(\frac{|\xi|^2}{2}\Bigr), $$

where $f_m$ is  a smooth one variable function $[0,\infty)\ni r\mapsto f_m(r)$.    

The function $f_m=f_{m,w}$ depends on the initial $w$. The dependence $w\mapsto f_{m,w}$ is linear and can be explicitly described in terms of the Hankel transforms.

Denote $\newcommand{\eF}{\mathscr{F}}$  by $\eF$ the class of  $C^2$-functions $f:[0,\infty)\to \bR$  such that

$$ f'(0) < f'(r),\;\; f'(0)< f'(r)+2rf''(r)<-f'(0),\;\;\forall r>0.  \tag{1} $$



Denote by $\newcommand{\eW}{\mathscr{W}}$ $\eW$  the collection of nonnegative    Schwartz functions $w :\bR\to \bR$  such that  

$$f_{m,w}\in\eF, \;\;\forall m>0. $$ 

 

**Remark 1.**  *The  class $\eF$  contains  all the completely  monotone functions $f:[0,\infty)\to \bR$ such that $f''(0)>0$.* (Recall that a function  $f:[0,\infty)\to \bR$ is completely monotone if it is smooth and  $(-1)^kf^{(k)}(t)\geq 0$, $\forall t\geq 0$, $\forall k\geq 0$.)  

This follows from the following observations.

 - The functions $r\mapsto g_s(r)= e^{-sr}$,  belong to $\eF$ for any $s>0$.
 - The  set $\eF$ is a convex cone.
 -  By Bernstein theorem, any completely monotone function  $f$  can be written as an infinite superposition of  nonnegative multiples of the functions $g_s$. More precisely, there exists  a finite positive Borel measure $\mu(|ds|)$ on $[0,\infty)$ such that

$$ f(r)=\int_0^\infty e^{-sr} \mu(|ds|). $$

**Remark 2.** *If  $w:\bR\to\bR$ is a nonnegative  Schwartz function  whose  restriction to $[0,\infty)$ is completely monotone, then $w\in \eW$*.

Indeed, we can find a positive, finite Borel measure $\mu(|ds|)$ on $[0,\infty)$ such that

$$ w(t)= \int_0^\infty e^{-st}\mu(|ds|). $$

Then

$$ w_m(x)=  \int_0^\infty e^{-s|x|^2/2}\mu(|ds|), $$

$$ V_m(\xi)= \int_0^\infty\left(\int_{\bR^m} e^{-\ii(x,\xi)} e^{-s|x|^2/2} dx\right) \mu(|ds|) $$

$$ = \int_0^\infty\left(\int_{\bR^m} e^{-\ii(y,\xi)/\sqrt{s}} e^{-|y|^2/2} dy\right) s^{-\frac{m}{2}}\mu(|ds|)  = (2\pi)^{\frac{m}{2}}\int_0^\infty e^{-\frac{|\xi|^2}{2s}} s^{-\frac{m}{2}}\mu(|ds|). $$

Hence

$$ f_m(r)= (2\pi)^{\frac{m}{2}}\int_0^\infty e^{-\frac{r}{s}} s^{-\frac{m}{2}}\mu(|ds|).$$

This proves that $f_m(r)$  is  completely monotone. A simple computation shows $f_m''(0)>0$  and from  **Remark 1**   we conclude $f_m\in\eF$.

Now comes the question.

> Does the class  $\eW$ contain examples of functions $w$ not covered by **Remark 2**?

**Remark 3.** I should  perhaps  mention  here a theorem of Schoenberg  which states that a function $w:[0,\infty)\to \bR$ is completely monotone if and only if, for any $m>0$ the function

$$ \bR^m\ni x\mapsto w_n(x)=w(|x|^2/2) \in\bR$$

is completely positive definite.  According to    Bochner's theorem, this means  that $w_m$  is the Fourier transform of a  positive measure on $\bR^m$.  In view of the above discussion we see  we can  rephrase the question as follows.

> Is it true that   $w\in \eW \Rightarrow f_{m,w}\geq 0$, $\forall m>0$?