# Explicit constructions of fundamental systems for spherical harmonics

For definitions and notations on the theory of spherical harmonics I refer to www.cis.upenn.edu/~cis610/sharmonics.pdf‎

Let $n,k\geq 0$, and let $S^n$ be the unit sphere on $\mathbb{C}^{n+1}$. Let $\mathcal{H}_k(S^n)$ the space of spherical harmonics that has dimension $a_{k,n+1}$. We say a set of unit vectors $\sigma_1,\ldots,\sigma_{a_{k,n+1}} \in S^n$ is a fundamental system if the matrix $(P_{k,n}(\sigma_i\cdot\sigma_j))_{i,j}$ is non-singular, where $P_{k,n}$ is the Gegenbauer polynomial of degree $k$ and dimension $n+1$.

Fundamental systems provide explicit parameterizations of $\mathcal{H}_k(S^n)$ in terms of basis functions $f_i(\tau)=P_{k,n}(\sigma_i\cdot\tau)$ on the unit sphere. In the notes above it is proved the existence of such systems for any $k,n\geq 1$ (Proposition 1.20).

My question is whether there are explicit constructions of these fundamental systems, or simple algorithms for obtaining such points.

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