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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)

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Convolution roots of radial positive definite functions with compact support
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by Werner Ehm, Tilmann Gneiting and Donald Richards PDF
Trans. Amer. Math. Soc. 356 (2004), 4655-4685 Request permission


A classical theorem of Boas, Kac, and Krein states that a characteristic function $\varphi$ with $\varphi (x) = 0$ for $|x| \geq \tau$ admits a representation of the form \[ \varphi (x) = \int \! u(y) \hspace {0.2mm} \overline {u(y+x)} \mathrm {d}y, \qquad x \in \mathbb {R}, \] where the convolution root $u \in L^2(\mathbb {R})$ is complex-valued with $u(x) = 0$ for $|x| \geq \tau /2$. 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 \[ \int |x|^2 f(x) \mathrm {d}x = - \Delta \varphi (0) \geq 4 j_{(d-2)/2}^2 \] 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
  • Email:
  • Tilmann Gneiting
  • Affiliation: Department of Statistics, University of Washington, Box 354322, Seattle, Washington 98195-4322
  • Email:
  • Donald Richards
  • Affiliation: Department of Statistics, Pennsylvania State University, 326 Thomas Building, University Park, Pennsylvania 16802-2111
  • MR Author ID: 190669
  • Email:
  • Received by editor(s): April 10, 2003
  • Received by editor(s) in revised form: September 2, 2003
  • Published electronically: May 10, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 4655-4685
  • MSC (2000): Primary 42A38, 42A82, 60E10
  • DOI:
  • MathSciNet review: 2067138