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  "body": "\n## Statement\n\nLet $d \\ge 3$ and $\\ell \\ge 0$ be integers.\nLet $\\mathcal{H}_\\ell$ be the space of [[spherical harmonics|3F32Y21HP4]] of degree $\\ell$ on the unit sphere $S^{d-1} \\subset \\R^d$, with dimension $N$.\nLet $\\sigma$ be the surface measure, $\\omega = \\sigma(S^{d-1})$, and $\\lambda = \\frac{d - 2}{2}$.\nLet $Y_1, \\dots, Y_N$ be an [[orthonormal basis|0S9GGTR0EX]] of $\\mathcal{H}_\\ell$ for the inner product $\\ip{f, g} = \\int_{S^{d-1}} f \\, \\overline{g} \\, d\\sigma$.\nThen for all $x, y \\in S^{d-1}$,\n$$\\sum_{j=1}^{N} Y_j(x) \\, \\overline{Y_j(y)} = \\frac{N}{\\omega} \\, \\frac{C_\\ell^{(\\lambda)}(x \\cdot y)}{C_\\ell^{(\\lambda)}(1)},$$\nwhere $C_\\ell^{(\\lambda)}$ is the [[Gegenbauer polynomial|5TRFA6F79B]].\nIn particular, for $d = 3$ and the standard basis $Y_\\ell^m$, $-\\ell \\le m \\le \\ell$,\n$$P_\\ell(x \\cdot y) = \\frac{4\\pi}{2\\ell + 1} \\sum_{m=-\\ell}^{\\ell} Y_\\ell^m(x) \\, \\overline{Y_\\ell^m(y)},$$\nwhere $P_\\ell$ is the [[Legendre polynomial|QFYGJVA996]].\n\n## Remarks\n\nProof idea: the left side $K(x, y)$ is the [[zonal spherical harmonic|Q5D54R3RGE]] $Z_y^{(\\ell)}(x)$, so it does not depend on the choice of basis.\nSince rotations act unitarily on $\\mathcal{H}_\\ell$, $K(gx, gy) = K(x, y)$ for every rotation $g$, so $K$ depends only on $x \\cdot y$.\nA harmonic of degree $\\ell$ invariant under the rotations fixing $y$ is a multiple of $C_\\ell^{(\\lambda)}(x \\cdot y)$.\nThe 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 $K(x, x) = N / \\omega$.\nExample: for $d = 3$ and $\\ell = 1$, the basis $\\sqrt{3/(4\\pi)} \\, x_i$, $i = 1, 2, 3$, gives $\\frac{3}{4\\pi} \\, x \\cdot y = \\frac{3}{4\\pi} P_1(x \\cdot y)$.\nSetting $x = y$ gives Unsöld's theorem: $\\sum_{m=-\\ell}^{\\ell} \\abs{Y_\\ell^m(x)}^2 = \\frac{2\\ell + 1}{4\\pi}$ at every point of $S^2$.\n\n## Explanations\n\n### L1 human\n\nPick two points on a sphere.\nSumming the products of all basic patterns of degree $\\ell$ at the two points gives a number that depends only on the angle between the points.\nThat number is a Legendre polynomial of the cosine of the angle, up to a fixed factor.\nSo a complicated sum over many functions collapses to a single simple function.\n\n### L2 human\n\nThe theorem identifies the reproducing kernel of $\\mathcal{H}_\\ell$ with a Gegenbauer polynomial in $x \\cdot y$.\nIt gives the projection of a function $f$ onto $\\mathcal{H}_\\ell$ as the convolution $\\frac{N}{\\omega} \\int_{S^{d-1}} f(y) \\frac{C_\\ell^{(\\lambda)}(x \\cdot y)}{C_\\ell^{(\\lambda)}(1)} \\, d\\sigma(y)$.\nCombined with the generating function of the Legendre polynomials, it expands $1/\\norm{x - y}$ in spherical harmonics, the standard multipole expansion of potential theory.\nIt also gives the bound $\\abs{Y(x)} \\le \\sqrt{N / \\omega} \\, \\norm{Y}_{L^2}$ for every $Y \\in \\mathcal{H}_\\ell$.\n",
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