Convolution roots of radial positive definite functions with compact support

A classical theorem of Boas, Kac, and Krein states that a characteristic function ψ with ψ(x) = 0 for |x| ≥ τ admits a representation of the form ψ (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 Boas-Kac representation under additional constraints: If ψ 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 ψ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on 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 ℝ^d whose characteristic function ψ vanishes outside the unit ball, then ∫ |x|² f (x) dx = -Δψ(0) ≥ 4j_(d-2)/2² 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.