Convolution Roots of Radial Positive Definite Functions with Compact Support

A classical theorem of Boas, Kac, and Krein states that a characteristic function φ with φ(x) = 0 for |x| ≥ τ admits a representation of the form φ(x) = ∫ u(y)u(y + x) dy, x ∈ R, where the convolution root u ∈ L2(R) is complex-valued with u(x) = 0 for |x| ≥ τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the BoasKac representation under additional constraints: If φ 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 φ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on Rd 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 Rd whose characteristic function φ vanishes outside the unit ball, then ∫ |x|f(x) dx = −∆φ(0) ≥ 4 j (d−2)/2 where jν denotes the first positive zero of the Bessel function Jν , 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 R2 does not exist.