## 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

## Abstract:

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.## References

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## Additional Information

**Werner Ehm**- Affiliation: Institut für Grenzgebiete der Psychologie und Psychohygiene, Wilhelmstrasse 3a, 79098 Freiburg, Germany
- Email: ehm@igpp.de
**Tilmann Gneiting**- Affiliation: Department of Statistics, University of Washington, Box 354322, Seattle, Washington 98195-4322
- Email: tilmann@stat.washington.edu
**Donald Richards**- Affiliation: Department of Statistics, Pennsylvania State University, 326 Thomas Building, University Park, Pennsylvania 16802-2111
- MR Author ID: 190669
- Email: richards@stat.psu.edu
- 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: https://doi.org/10.1090/S0002-9947-04-03502-0
- MathSciNet review: 2067138