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

Abstract

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