I know that a Fourier series of a function that is differentiable will converge to that function. If the function isn't differentiable say it exhibits a jump discontinuity - the Fourier series may not converge, and could instead oscillate about the true value. One example of this effect is the Gibbs phenomenon. Spherical harmonics function a lot like 2d Fourier series. Do they exhibit similar ringing when used to represent discontinuous angular functions?
2026-03-25 22:03:50.1774476230
Are spherical harmonics subject to the Gibbs phenomenon?
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The answer is yes. Just like for the Fourier series on the torus, spherical harmonics decompositions on the sphere also exhibit Gibbs oscillations when there are discontinuities.
This has been documented and methods for removing them have been studied. For instance, look at this.