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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Integral representation of functions and distributions positive definite relative to the orthogonal group
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by A. E. Nussbaum PDF
Trans. Amer. Math. Soc. 175 (1973), 355-387 Request permission


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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 175 (1973), 355-387
  • MSC: Primary 43A35
  • DOI:
  • MathSciNet review: 0333600