Let $f:\mathbb{R}^d\to\mathbb{R}$ be a smooth compactly supported covariance function of a stationary random fields (hence positive definite).

Is there a compactly supported function $g:\mathbb{R}^d\to \mathbb{C}$ such that $g\ast g=f$ ?

Using Fourier transform and Bochner's theorem, this question is equivalent to the construction of a entire square root for entire function of exponential type that are positive and fast decaying on the real line.