[*power growth at infinity*]{}, i.e. there exists a number $ \beta>0$ so that $$\lim_{|x|_2\to \infty} \frac{f(x)} {|x|_2^\beta}=0,$$ where $|\cdot|_2$ stands for the Euclidean norm on $\R^n.$ Then $f$ represents a distribution acting by integration: for every $\phi\in {\mathcal S},$ $$\langle f, \phi \rangle = \int_{\R^n} f(x) \phi(x)\ dx.$$

The Fourier transform of a distribution $f$ is defined by $\langle\hat{f}, \phi\rangle= \langle f, \hat{\phi} \rangle$ for every test function $\phi.$ If $\phi$ is an even test function, then $(\hat{\phi})^\wedge = (2\pi)^n \phi$, so the Fourier transform is self-invertible (up to a constant) for even distributions.

A distribution $f$ is called even homogeneous of degree $p\in \R$ if $$\langle f(x), \phi(x/\alpha) \rangle = |\alpha|^{n+p} 
\langle f,\phi \rangle$$ for every test function $\phi$ and every $\alpha\in \R,\ \alpha\neq 0.$ The Fourier transform of an even homogeneous distribution of degree $p$ is an even homogeneous distribution of degree $-n-p.$

We say that a distribution is [*positive definite*]{} if its Fourier transform is a positive distribution in the sense that $\langle \hat{f},\phi \rangle \ge 0$ for every non-negative test function $\phi.$ Schwartz’s generalization of Bochner’s theorem (see, for example, [@GV p.152]) states that a distribution is positive definite if and only if it is the 