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 \[ φ (x) = ∫ \! u(y) {0.2mm} u(y+x)̄ dy, x ∈ R, \] 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 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 \[ ∫ |x|^2 f(x) dx = - Δ φ (0) ≥ 4 j(d-2)/2^2 \] where j_ν denotes the first positive zero of the Bessel function J_ν, 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.