For each (v>0) and (omega in mathbb{R}^{3}), the Matrn function (M_{v}(|x-x|) e^{i omega^{prime}left(x-x^{prime}ight)}) defines a stationary complex

For each \(v>0\) and \(\omega \in \mathbb{R}^{3}\), the Matérn function \(M_{v}(\|x-x\|) e^{i \omega^{\prime}\left(x-x^{\prime}ight)}\) defines a stationary complex Gaussian process on \(\mathbb{R}^{3}\) with frequency \(\|\omega\|\) and wave direction \(\omega /\|\omega\|\). For each \(ho \in \mathbb{R}^{3}\) such that \(\|ho\| \leq 1\), deduce that the following functions are positive definite symmetric on \(\mathbb{R}^{3} \times[3]\) :

\[
\begin{aligned}
& M_{v}\left(\left\|x-x^{\prime}ight\|ight) \delta_{r s} \\
& M_{v}\left(\left\|x-x^{\prime}ight\|ight) \cos \left(\omega^{\prime}\left(x-x^{\prime}ight)ight) \delta_{r s} \\
& M_{v}\left(\left\|x-x^{\prime}ight\|ight) \cos \left(\omega^{\prime}\left(x-x^{\prime}ight)ight) \delta_{r s}-M_{v}\left(\left\|x-x^{\prime}ight\|ight) \sin \left(\omega^{\prime}\left(x-x^{\prime}ight)ight) \chi(ho)_{r s}
\end{aligned}
\]

for \(x, x^{\prime} \in \mathbb{R}^{3}\) and \(r, s \in[3]\).