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Integral representations of positive definite matrix-valued distributions on cylinders


Author: Jürgen Friedrich
Journal: Trans. Amer. Math. Soc. 313 (1989), 275-299
MSC: Primary 43A35; Secondary 46F25
DOI: https://doi.org/10.1090/S0002-9947-1989-0992599-1
MathSciNet review: 992599
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Abstract: The notion of a $ G$-continuous matrix-valued positive definite distribution on

$\displaystyle {S_N}(2a) \times {{\mathbf{R}}^M} \times G$

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.

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DOI: https://doi.org/10.1090/S0002-9947-1989-0992599-1
Article copyright: © Copyright 1989 American Mathematical Society

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