For each \(ho \in \mathbb{R}^{3}\) such that \(\|ho\| \leq 1\), deduce that the following symmetric functions are positive definite on \(\mathbb{R} \times[3]\) :

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