Integral representations of positive definite matrix-valued distributions on cylinders

Abstract:

The notion of a $G$-continuous matrix-valued positive definite distribution on \[ {S_N}(2a) \times {{\mathbf {R}}^M} \times G\] is introduced, where $G$ is an abelian separable locally compact group and where ${S_N}(2a)$ is an open ball around zero in ${\mathbf {R}^N}$ with radius $2a > 0$. This notion generalizes that one of strongly continuous positive definite operator-valued functions. For these objects, a Bochner-type theorem gives a suitable integral representation if $N = 1$ or if the matrix-valued distribution is invariant w.r.t. rotations in ${\mathbf {R}^N}$. As a consequence, appropriate extensions to the whole group are obtained. In particular, we show that a positive definite function on a certain cylinder in a separable real Hilbert space $H$ may be extended to a characteristic function of a finite positive measure on $H$, if it is invariant w.r.t. rotations and continuous w.r.t. a suitable topology.