On functions positive definite relative to the orthogonal group and the representation of functions as Hankel-Stieltjes transforms

Abstract:

To every continuous function $f$ on an interval $0 \leq x < a(0 < a \leq \infty )$ and every positive number $\nu$ associate the kernel \[ f(x,y) = \int _0^\pi f\left ( (x^2 + y^2 - 2xy \cos \theta )^{1/2}\right ) (\sin \theta )^{2\nu - 1},d\theta ,\quad 0 < x, y < a/2. \] Let $\Omega (z) = \Gamma (\nu + 1/2) (2/z)^{\nu - 1/2} J_{\nu -1/2} J_{\nu - 1/2}(z)$, where $J_{\nu - 1/2}(z)$ is the Bessel function of index $\nu - 1/2$. It is shown that $f$ has an integral representation $f(x) = \int _{-\infty }^\infty \Omega (x\sqrt \lambda )d\gamma (\lambda )$, where $\gamma$ is a finite, positive Radon measure on $R$, if and only if the kernel $f(x,y)$ is positive definite. If $\nu = (N - 1)/2$, where $N$ is an integer $\geq 2$, this condition is equivalent to ${f_N}(x) = f(|x|),\;x \in {R^N},\;|x| < \alpha$, is positive definite relative to the orthogonal group $O(N)$. The results of this investigation extend the preceding one of the author on functions positive definite relative to the orthogonal group. In particular they yield the result of Rudin on the extensions of radial positive definite functions.