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We study the expansions of the elements in $\mathcal S(\mathbb{R}_+^d)$ and$\mathcal{S}'(\mathbb{R}_+^d)$ with respect to the Laguerre orthonormal basis,extending the result of M. Guillemont-Teissier in the case $d=1$. As aconsequence, we obtain the Schwartz kernel theorem for$\mathcal{S}(\mathbb{R}_+^d)$ and $\mathcal{S}'(\mathbb{R}_+^d)$ and theextension theorem of Whitney type for $\mathcal{S}(\mathbb{R}_+^d)$.

Extension theorem of Whitney type for $\mathcal S(\mathbb{R}_+^d)$ by the use of the Kernel Theorem