= Harmonic decomposition of homogeneous polynomials
{title2=$\mathcal P_\ell=\mathcal H_\ell\oplus|x|^2\mathcal P_{\ell-2}$}
= Fischer decomposition
{c}
{synonym}
If $\mathcal P_\ell$ is the space of degree-$\ell$ <homogeneous polynomials> and $\mathcal H_\ell$ its <harmonic polynomials>, then $\mathcal P_\ell=\mathcal H_\ell\oplus|x|^2\mathcal P_{\ell-2}$. Under the <Fischer inner product>, the second summand is orthogonal to the kernel of the <Laplacian>. Iteration expresses every polynomial restricted to the unit <sphere> as a sum of <spherical harmonics>.
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