|
5
3
|
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. |
||||||
|
|
5
|
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. |
|||||||||
|
|
-1
|
can anyone help me.. What is the minimun finite subset $X$ of $S^2$ such that for any harmonic polynomial of degree 4 we have $\displaystyle\sum_{x\in X}{f(x)}=0$? |
|||||
|
