Addition theorem for spherical harmonics

Remarks

Proof idea: the left side $K(x, y)$ is the zonal spherical harmonic $Z_y^{(\ell)}(x)$, so it does not depend on the choice of basis. Since rotations act unitarily on $\mathcal{H}_\ell$, $K(gx, gy) = K(x, y)$ for every rotation $g$, so $K$ depends only on $x \cdot y$. A harmonic of degree $\ell$ invariant under the rotations fixing $y$ is a multiple of $C_\ell^{(\lambda)}(x \cdot y)$. The constant comes from $K(x, x)$ being constant and $\int_{S^{d-1}} K(x, x) \, d\sigma(x) = \sum_j \norm{Y_j}^2 = N$, so $K(x, x) = N / \omega$. Example: for $d = 3$ and $\ell = 1$, the basis $\sqrt{3/(4\pi)} \, x_i$, $i = 1, 2, 3$, gives $\frac{3}{4\pi} \, x \cdot y = \frac{3}{4\pi} P_1(x \cdot y)$. Setting $x = y$ gives Unsöld's theorem: $\sum_{m=-\ell}^{\ell} \abs{Y_\ell^m(x)}^2 = \frac{2\ell + 1}{4\pi}$ at every point of $S^2$.

Going deeper

For a student

The theorem identifies the reproducing kernel of $\mathcal{H}_\ell$ with a Gegenbauer polynomial in $x \cdot y$. It gives the projection of a function $f$ onto $\mathcal{H}_\ell$ as the convolution $\frac{N}{\omega} \int_{S^{d-1}} f(y) \frac{C_\ell^{(\lambda)}(x \cdot y)}{C_\ell^{(\lambda)}(1)} \, d\sigma(y)$. Combined with the generating function of the Legendre polynomials, it expands $1/\norm{x - y}$ in spherical harmonics, the standard multipole expansion of potential theory. It also gives the bound $\abs{Y(x)} \le \sqrt{N / \omega} \, \norm{Y}_{L^2}$ for every $Y \in \mathcal{H}_\ell$.