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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 $g:\mathbb{R}^d\to \mathbb{C}$ (not necessarily positive definite) 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.

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 $g:\mathbb{R}^d\to \mathbb{C}$ (not necessarily positive definite) 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.

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.

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Chris
Chris
  • 321
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Particular Ehrenpreis factorization for covariance function

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 $g:\mathbb{R}^d\to \mathbb{C}$ (not necessarily positive definite) 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.