Hello,
Does someone know an explicit basis of the space of harmonic homogeneous polynomial in N variables.
When $N=3$, if I'm not mistaking Legendre polynomial allow to write an explicit basis.
Is there a known explicit basis when $N > 3$ ?
Thanks for your answers, and reference in case you know one.
Let $K$ denote the Kelvin transform, and let $|\alpha|:=\sum_{j=1}^n\alpha_j$ denote the weight of the multi-index $\alpha\in\mathbb{N}^n$. Then, an explicit base for the space of homogeneous harmonic polynomials in $n$ variables and degreee $m$, $\mathcal H^m:=\mathcal H^m[x_1,\dots ,x_n]$, is $$\big\{ K\big(\partial^\alpha\|x\|^{2-n}\big) : |\alpha|=m, \alpha_n\leq 1 \big\}.$$ Indeed, an easy combinatorial computation shows it has the right cardinality $$\operatorname{card}\{\alpha\in\mathbb{N}^n : |\alpha|=m, \alpha_n\leq 1 \}= {n+m-1\choose n-1} - {n+m-3\choose n-1} = \operatorname{dim}\mathcal H^m,$$ (the latter dimension being already known from linear algebra considerations on the operator $\Delta$). On the other hand, one can verify that it spans the same linear space as the analogous set without the constraint on $\alpha_n$: $$\{ K\big(\partial^\alpha\|x\|^{2-n}\big): |\alpha|=m\},$$ which is the whole space $\mathcal H^m$. So it is a base.
A nice (and free) reference for these classical facts is Harmonic Function Theory, by S.Axler, P.Bourdon, and W.Ramey; see thm 5.25.
