I'd like to find a (hopefully modern, mathematician-friendly) reference which proves that homogeneous harmonic polynomials restrict to an orthonormal basis for $L^2$ functions on the sphere $S^n$. Particularly I'd like to understand why we should expect that harmonic functions on $\mathbb{R}^{n+1}$ restrict to eigenfunctions of the Laplacian on the sphere.
2026-03-26 20:43:32.1774557812
Reference request on spherical harmonics
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Chapter 5 of Harmonic Function Theory by Axler, Bourdon and Ramey. As with any Axler book, it's a very thorough treatment of harmonic polynomials with a fair bit of motivation.