Questions tagged [harmonic-polynomials]

This MO answer cites the Goodman-Wallach book to affirm that: $$\mathrm{Sym}^k\left(\mathbb{R}^n\right)=\mathcal{H}^k\oplus q\mathcal{H}^{k-2}\oplus q^2\mathcal{H}^{k-4}\oplus\cdots$$ with $\mathrm{...

Preamble $\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that If $G$ is a subgroup of $\SO(m+1)$ ...

For polynomials $f,g \in \mathbb{C}[x_1,\dots,x_n]$, the following inner product appears frequently in the literature of harmonic polynomials: $$ \langle f,g \rangle = f(\partial/\partial x_1, \dots, \...