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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A short proof of Schoenberg's theorem


Author: Paul Ressel
Journal: Proc. Amer. Math. Soc. 57 (1976), 66-68
MSC: Primary 44A10
DOI: https://doi.org/10.1090/S0002-9939-1976-0405007-2
MathSciNet review: 0405007
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Abstract: Using positive semidefiniteness of Laplace transforms, we give a short and simple proof of Schoenberg's theorem characterising radially symmetric positive semidefinite functions on a Hilbert space. A slight generalisation of this theorem is also given.


References [Enhancements On Off] (What's this?)

  • [1] W. F. Donoghue, Distributions and Fourier transforms, Academic Press, New York, 1969.
  • [2] Paul Ressel, Laplace-Transformation nichtnegativer und vektorwertiger Maße, Manuscripta Math. 13 (1974), 143–152 (German, with English summary). MR 0344886, https://doi.org/10.1007/BF01411492
  • [3] I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. of Math. 39 (1938).

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DOI: https://doi.org/10.1090/S0002-9939-1976-0405007-2
Article copyright: © Copyright 1976 American Mathematical Society