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Let $p \geq 2$ be an integer and denote by $\mathcal{S}(\mathbb{R}^p)$ the space of Schwartz functions on $\mathbb{R}^p$. Let $\mathcal{S}(\mathbb{R}^p)^{O(p)}$ be the subspace of rotation-invariant Schwartz functions. For each integer $m \geq 0$, let $\mathcal{H}_{p,m}$ denote the finite dimensional space of polynomial, harmonic functions $u: \mathbb{R}^p \rightarrow \mathbb{C}$ that are homogenous of degree $m$. Denote by $\mathcal{H}_{p,m} \otimes \mathcal{S}(\mathbb{R}^p)^{O(p)}$ the linear span of all Schwartz functions $x \mapsto u(x) r(x)$ where $u \in \mathcal{H}_{p,m}$ and $r \in \mathcal{S}(\mathbb{R}^p)^{O(p)}$.

Is it true that the algebraic sum of subspaces $\sum_{m \geq 0}{\mathcal{H}_{p,m} \otimes \mathcal{S}(\mathbb{R}^p)^{O(p)}}$ is dense in $\mathcal{S}(\mathbb{R}^p)$, with respect to the Schwartz topology?

This decomposition is probably "well-known" in the $L^2$-sense (see for example Stein-Weiss, chapter 3), but I am particularly interested in the Schwartz-topology case. There are also some exercises related to this in chapter 3 of Howe-Tan's book "Non-abelian harmonic analysis", but I'd be interested in a reference that deals with the Schwartz topology (and the statements are not perhaps implicit in exercises.)

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Yes.

You can use density of Hermite functions, see https://aip.scitation.org/doi/abs/10.1063/1.1665472 then you have that functions $e^{-x^2}P(x)$, with $P$ a multivariate polynomial, are dense. Finally decompose the space of such polynomials into irreducible representations of the orthogonal group.

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