= Spectrum of the round sphere
{title2=$\lambda_n=n(n+d-1)$}
For the positive <Laplace-Beltrami operator> on the unit <sphere> $S^d$, with $d\geq1$, the <eigenvalues> are $n(n+d-1)$ for integers $n\geq0$. Their <eigenspaces> consist of restrictions of degree-$n$ <harmonic polynomials> and have dimensions $\binom{n+d}{d}-\binom{n+d-2}{d}$, with the second term zero for $n<2$. The <Laplacian in polar coordinates> gives the eigenvalue; <Fischer decomposition> and the <Stone-Weierstrass theorem> prove completeness.
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