Integral representation of functions and distributions positive definite relative to the orthogonal group

Abstract:

A continuous function f on an open ball B in ${R^N}$ is called positive definite relative to the orthogonal group $O(N)$ if f is radial and $\smallint \smallint f(x - y)\phi (x)\overline {\phi (y)} \;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. It is shown that f is positive definite in B relative to $O(N)$ if and only if f has an integral representation $f(x) = \smallint {e^{ix \cdot t}}d{\mu _1}(t) + \smallint {e^{x \cdot t}}d{\mu _2}(t)$, where ${\mu _1}$ and ${\mu _2}$ are bounded, positive, rotation invariant Radon measures on ${R^N}$ and ${\mu _2}$ may be taken to be zero if, in addition to f being positive definite relative to $O(N),\smallint \smallint f(x - y)( - \Delta \phi )(x)\phi (y)\;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. Both conditions are satisfied if f is a radial positive definite function in B. Thus the theorem yields as a special case Rudin’s theorem on the extension of radial positive definite functions. The result is extended further to distributions.