# Are radial Schwartz functions tensored harmonic polynomials dense in the Schwartz space?

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.)

## 1 Answer

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.