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We propose an algorithm for robust recovery of the spherical harmonicexpansion of functions defined on the d-dimensional unit sphere$\mathbb{S}^{d-1}$ using a near-optimal number of function evaluations. We showthat for any $f \in L^2(\mathbb{S}^{d-1})$, the number of evaluations of $f$needed to recover its degree-$q$ spherical harmonic expansion equals thedimension of the space of spherical harmonics of degree at most $q$ up to alogarithmic factor. Moreover, we develop a simple yet efficient algorithm torecover degree-$q$ expansion of $f$ by only evaluating the function onuniformly sampled points on $\mathbb{S}^{d-1}$. Our algorithm is based on theconnections between spherical harmonics and Gegenbauer polynomials and leveragescore sampling methods. Unlike the prior results on fast spherical harmonictransform, our proposed algorithm works efficiently using a nearly optimalnumber of samples in any dimension d. We further illustrate the empiricalperformance of our algorithm on numerical examples.

Near Optimal Reconstruction of Spherical Harmonic Expansions