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Transactions of the American Mathematical Society

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

 

 

On functions positive definite relative to the orthogonal group and the representation of functions as Hankel-Stieltjes transforms


Author: A. Edward Nussbaum
Journal: Trans. Amer. Math. Soc. 175 (1973), 389-408
MSC: Primary 43A70; Secondary 44A15
MathSciNet review: 0333601
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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

$\displaystyle f(x,y) = \int_0^\pi {f({{({x^2} + {y^2} - 2xy\;\cos \;\theta )}^{1/2}}){{(\sin \;\theta )}^{2\nu - 1}}d\theta ,\quad 0 < x,y < a/2.} $

Let $ \Omega (z) = \Gamma (\nu + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\ke... ...iptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}(z)$, where $ {J_{\nu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}(z)$ is the Bessel function of index $ \nu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. It is shown that f has an integral representation $ f(x) = \smallint_{ - \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(\vert x\vert),\;x \in {R^N},\;\vert x\vert < \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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DOI: https://doi.org/10.1090/S0002-9947-1973-0333601-8
Keywords: Positive definite functions, integral representation, Hankel-Stieltjes transform
Article copyright: © Copyright 1973 American Mathematical Society