Fischer decomposition

Remarks

Proof idea: on the space $P_m$ of homogeneous polynomials of degree $m$, use the inner product $\langle p, q \rangle = p(\partial) \bar q$, where $p(\partial)$ replaces each $x_i$ by $\partial / \partial x_i$. For this inner product, multiplication by $\abs{x}^2$ from $P_{m-2}$ to $P_m$ is the adjoint of the Laplacian from $P_m$ to $P_{m-2}$. So the harmonic polynomials of degree $m$, which form the kernel of the Laplacian, are the orthogonal complement of $\abs{x}^2 P_{m-2}$ in $P_m$. Example: $x_1^2 = \left(x_1^2 - \tfrac{1}{n}\abs{x}^2\right) + \abs{x}^2 \cdot \tfrac{1}{n}$, and the first term is harmonic since its Laplacian is $2 - \tfrac{2n}{n} = 0$. Restricted to the unit sphere, where $\abs{x}^2 = 1$, the expansion shows that every polynomial is a finite sum of spherical harmonics.

Going deeper

For a student

The decomposition gives the dimension of the space of homogeneous harmonic polynomials of degree $m$: it is $\dim P_m - \dim P_{m-2}$, which is $2m + 1$ in $\R^3$. It shows that the restrictions of polynomials to the sphere are dense in the continuous functions on the sphere, and that they are sums of spherical harmonics. This leads to the decomposition of $L^2$ of the sphere into spherical harmonics, which underlies the expansion in spherical harmonics.