Integral representations of positive definite matrix-valued distributions on cylinders

Abstract

The notion of a G G -continuous matrix-valued positive definite distribution on \[ S N ( 2 a ) × R M × G {S_N}(2a) \times {{\mathbf {R}}^M} \times G \] is introduced, where G G is an abelian separable locally compact group and where S N ( 2 a ) {S_N}(2a) is an open ball around zero in R N {\mathbf {R}^N} with radius 2 a > 0 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 N = 1 or if the matrix-valued distribution is invariant w.r.t. rotations in R N {\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 H may be extended to a characteristic function of a finite positive measure on H H , if it is invariant w.r.t. rotations and continuous w.r.t. a suitable topology.