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

TL;DR

This paper extends Whitney's extension theorem for the space of Schwartz functions on the positive real orthant using Laguerre basis expansions and the Kernel Theorem, generalizing previous one-dimensional results.

Contribution

It generalizes Whitney's extension theorem for Schwartz spaces on b, utilizing Laguerre basis expansions and the Kernel Theorem to extend prior one-dimensional findings.

Findings

Established Laguerre basis expansions for b spaces

Derived Schwartz kernel theorem for b spaces

Extended Whitney's extension theorem to b setting

Abstract

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 a consequence, we obtain the Schwartz kernel theorem for $\mathcal{S}(\mathbb{R}_+^d)$ and $\mathcal{S}'(\mathbb{R}_+^d)$ and the extension theorem of Whitney type for $\mathcal{S}(\mathbb{R}_+^d)$.