Zonal spherical harmonics

Remarks

It exists and is unique because $Y \mapsto Y(y)$ is a linear functional on the finite-dimensional inner product space $\mathcal{H}_\ell$. If $Y_1, \dots, Y_N$ is an orthonormal basis of $\mathcal{H}_\ell$, then $Z_y^{(\ell)}(x) = \sum_{j=1}^{N} Y_j(x) \overline{Y_j(y)}$. It is real-valued, and $Z_{gy}^{(\ell)}(gx) = Z_y^{(\ell)}(x)$ for every rotation $g$, so it depends only on $x \cdot y$. Example: for $d = 3$ and $\ell = 1$, $Z_y^{(1)}(x) = \frac{3}{4\pi} \, x \cdot y$. Check: 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 $a \cdot y = Y(y)$. Non-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 $Y(y)$. In 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.