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Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive definite for $0<\gamma<1$?

Any hints will be greatly appreciated. I have found the following Theorem: A continuous function $\varphi:[0, \infty) \rightarrow \mathbb{R}$ is positive definite and radial on $\mathbb{R}^s$ for all $s$ if and only if it is of the form $$ \varphi(r)=\int_0^{\infty} e^{-r^2 t^2} d \mu(t), $$ where $\mu$ is a finite non-negative Borel measure on $[0, \infty)$. $ e^{-r^{2\gamma} t^2}$ is positive definite but I do not know how that helps me.

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive definite for $0<\gamma<1$?

Any hints will be greatly appreciated. I have found the following Theorem: A continuous function $\varphi:[0, \infty) \rightarrow \mathbb{R}$ is positive definite and radial on $\mathbb{R}^s$ for all $s$ if and only if it is of the form $$ \varphi(r)=\int_0^{\infty} e^{-r^2 t^2} d \mu(t), $$ where $\mu$ is a finite non-negative Borel measure on $[0, \infty)$. $ e^{-r^{2\gamma} t^2}$ is positive definite but I do not know how that helps me. UPDATE: the problem with this theorem, the function has to be positive definite/radial for ALL $s$. At the moment I can only show that my function is positive definite for $s=1$. I suspect it is positive definite for $s=2,3$. Are there other results for specific $s$?

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive definite for $0<\gamma<1$?

Any hints will be greatly appreciated.

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive definite for $0<\gamma<1$?

Any hints will be greatly appreciated. I have found the following Theorem: A continuous function $\varphi:[0, \infty) \rightarrow \mathbb{R}$ is positive definite and radial on $\mathbb{R}^s$ for all $s$ if and only if it is of the form $$ \varphi(r)=\int_0^{\infty} e^{-r^2 t^2} d \mu(t), $$ where $\mu$ is a finite non-negative Borel measure on $[0, \infty)$. $ e^{-r^{2\gamma} t^2}$ is positive definite but I do not know how that helps me.

Let $f(x)$ be a positive definite function on $x \in R$. Assume $f(x)$ is isotropic , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive definite for $0<\gamma<1$? I know how to prove it for a completely monotone function and I have a feeling I saw a proof for a radial positive definite function somewhere. Any hints will be greatly appreciated. Ideally I need this for a general $x \in R^d$

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive definite for $0<\gamma<1$? 

Any hints will be greatly appreciated.