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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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On functions positive definite relative to the orthogonal group and the representation of functions as Hankel-Stieltjes transforms
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by A. Edward Nussbaum PDF
Trans. Amer. Math. Soc. 175 (1973), 389-408 Request permission

Abstract:

To every continuous function $f$ on an interval $0 \leq x < a(0 < a \leq \infty )$ and every positive number $\nu$ associate the kernel \[ f(x,y) = \int _0^\pi f\left ( (x^2 + y^2 - 2xy \cos \theta )^{1/2}\right ) (\sin \theta )^{2\nu - 1},d\theta ,\quad 0 < x, y < a/2. \] Let $\Omega (z) = \Gamma (\nu + 1/2) (2/z)^{\nu - 1/2} J_{\nu -1/2} J_{\nu - 1/2}(z)$, where $J_{\nu - 1/2}(z)$ is the Bessel function of index $\nu - 1/2$. It is shown that $f$ has an integral representation $f(x) = \int _{-\infty }^\infty \Omega (x\sqrt \lambda )d\gamma (\lambda )$, where $\gamma$ is a finite, positive Radon measure on $R$, if and only if the kernel $f(x,y)$ is positive definite. If $\nu = (N - 1)/2$, where $N$ is an integer $\geq 2$, this condition is equivalent to ${f_N}(x) = f(|x|),\;x \in {R^N},\;|x| < \alpha$, is positive definite relative to the orthogonal group $O(N)$. The results of this investigation extend the preceding one of the author on functions positive definite relative to the orthogonal group. In particular they yield the result of Rudin on the extensions of radial positive definite functions.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 175 (1973), 389-408
  • MSC: Primary 43A70; Secondary 44A15
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0333601-8
  • MathSciNet review: 0333601