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  "body": "\n## Definition\n\nLet $d \\ge 2$ and $\\ell \\ge 0$ be integers, and let $\\sigma$ be the surface measure on the unit sphere $S^{d-1} \\subset \\R^d$.\nLet $\\mathcal{H}_\\ell$ be the space of [[spherical harmonics|3F32Y21HP4]] of degree $\\ell$ on $S^{d-1}$.\nEquip $\\mathcal{H}_\\ell$ with the [[inner product|0NB0AWGJBW]] $\\ip{f, g} = \\int_{S^{d-1}} f \\, \\overline{g} \\, d\\sigma$.\nLet $y \\in S^{d-1}$.\nThe *zonal spherical harmonic* of degree $\\ell$ with pole $y$ is the unique $Z_y^{(\\ell)} \\in \\mathcal{H}_\\ell$ such that\n$$Y(y) = \\int_{S^{d-1}} Y(x) \\, \\overline{Z_y^{(\\ell)}(x)} \\, d\\sigma(x) \\quad \\text{for all } Y \\in \\mathcal{H}_\\ell.$$\n\n## Remarks\n\nIt exists and is unique because $Y \\mapsto Y(y)$ is a linear functional on the finite-dimensional inner product space $\\mathcal{H}_\\ell$.\nIf $Y_1, \\dots, Y_N$ is an orthonormal basis of $\\mathcal{H}_\\ell$, then $Z_y^{(\\ell)}(x) = \\sum_{j=1}^{N} Y_j(x) \\overline{Y_j(y)}$.\nIt is real-valued, and $Z_{gy}^{(\\ell)}(gx) = Z_y^{(\\ell)}(x)$ for every rotation $g$, so it depends only on $x \\cdot y$.\nExample: for $d = 3$ and $\\ell = 1$, $Z_y^{(1)}(x) = \\frac{3}{4\\pi} \\, x \\cdot y$.\nCheck: every $Y \\in \\mathcal{H}_1$ has the form $Y(x) = x \\cdot a$ with $a \\in \\C^3$, and $\\int_{S^2} (x \\cdot a)(x \\cdot y) \\, d\\sigma(x) = \\frac{4\\pi}{3} \\, a \\cdot y$, so the integral returns $a \\cdot y = Y(y)$.\nNon-example: the function $x \\mapsto x \\cdot y$ itself is zonal and lies in $\\mathcal{H}_1$, but it fails the reproducing identity, since it returns $\\frac{4\\pi}{3} Y(y)$ instead of $Y(y)$.\nIn general $Z_y^{(\\ell)}(x) = \\frac{N}{\\sigma(S^{d-1})} \\frac{C_\\ell^{(\\lambda)}(x \\cdot y)}{C_\\ell^{(\\lambda)}(1)}$ for $d \\ge 3$, with $\\lambda = \\frac{d - 2}{2}$, $N = \\dim \\mathcal{H}_\\ell$ and $C_\\ell^{(\\lambda)}$ the [[Gegenbauer polynomial|5TRFA6F79B]].\n",
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