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Transactions of the American Mathematical Society

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Convolution roots of radial positive definite functions with compact support

Authors: Werner Ehm, Tilmann Gneiting and Donald Richards
Journal: Trans. Amer. Math. Soc. 356 (2004), 4655-4685
MSC (2000): Primary 42A38, 42A82, 60E10
Published electronically: May 10, 2004
MathSciNet review: 2067138
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Abstract: A classical theorem of Boas, Kac, and Krein states that a characteristic function $\varphi$ with $\varphi(x) = 0$ for $\vert x\vert \geq \tau$ admits a representation of the form

\begin{displaymath}\varphi(x) = \int u(y) \hspace{0.2mm} \overline{u(y+x)} \, \mathrm{d}y, \qquad x \in \mathbb{R}, \end{displaymath}

where the convolution root $u \in L^2(\mathbb{R})$ is complex-valued with $u(x) = 0$ for $\vert x\vert \geq \tau/2$. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If $\varphi$ is real-valued and even, can the convolution root $u$ be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of $\varphi$ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on $\mathbb{R}^d$ is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if $f$ is a probability density on $\mathbb{R}^d$ whose characteristic function $\varphi$ vanishes outside the unit ball, then

\begin{displaymath}\int \vert x\vert^2 f(x) \, \mathrm{d}x = - \Delta \varphi(0) \geq 4 \, j_{(d-2)/2}^2 \end{displaymath}

where $j_\nu$ denotes the first positive zero of the Bessel function $J_\nu$, and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in $\mathbb{R}^2$ does not exist.

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Additional Information

Werner Ehm
Affiliation: Institut für Grenzgebiete der Psychologie und Psychohygiene, Wilhelmstrasse 3a, 79098 Freiburg, Germany

Tilmann Gneiting
Affiliation: Department of Statistics, University of Washington, Box 354322, Seattle, Washington 98195-4322

Donald Richards
Affiliation: Department of Statistics, Pennsylvania State University, 326 Thomas Building, University Park, Pennsylvania 16802-2111

Received by editor(s): April 10, 2003
Received by editor(s) in revised form: September 2, 2003
Published electronically: May 10, 2004
Article copyright: © Copyright 2004 American Mathematical Society