Convolution roots of radial positive definite functions with compact support

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.