Integral Representations of Positive Definite Matrix-valued Distributions on Cylinders

The notion of a <7-continuous matrix-valued positive definite distribution on 5/v(2a) x RM x G is introduced, where G is an abelian separable locally compact group and where Sp/(2a) is an open ball around zero in RA' with radius la > 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 = \ or if the matrix-valued distribution is invariant w.r.t. rotations in R^ . 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 Hubert 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.