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Integral representation of functions and distributions positive definite relative to the orthogonal group

Author: A. E. Nussbaum
Journal: Trans. Amer. Math. Soc. 175 (1973), 355-387
MSC: Primary 43A35
MathSciNet review: 0333600
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Abstract: A continuous function f on an open ball B in $ {R^N}$ is called positive definite relative to the orthogonal group $ O(N)$ if f is radial and $ \smallint \smallint f(x - y)\phi (x)\overline {\phi (y)} \;dx\;dy \geq 0$ for all radial $ \phi \in C_0^\infty (B/2)$. It is shown that f is positive definite in B relative to $ O(N)$ if and only if f has an integral representation $ f(x) = \smallint {e^{ix \cdot t}}d{\mu _1}(t) + \smallint {e^{x \cdot t}}d{\mu _2}(t)$, where $ {\mu _1}$ and $ {\mu _2}$ are bounded, positive, rotation invariant Radon measures on $ {R^N}$ and $ {\mu _2}$ may be taken to be zero if, in addition to f being positive definite relative to $ O(N),\smallint \smallint f(x - y)( - \Delta \phi )(x)\phi (y)\;dx\;dy \geq 0$ for all radial $ \phi \in C_0^\infty (B/2)$. Both conditions are satisfied if f is a radial positive definite function in B. Thus the theorem yields as a special case Rudin's theorem on the extension of radial positive definite functions. The result is extended further to distributions.

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Keywords: Positive definite functions, positive definite distributions, expansions into generalized eigenvectors, nuclear spectral theorem
Article copyright: © Copyright 1973 American Mathematical Society

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