Spherical Harmonics - Spherical Harmonics Expansion

Spherical Harmonics Expansion

The Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions. On the unit sphere, any square-integrable function can thus be expanded as a linear combination of these:

This expansion holds in the sense of mean-square convergence — convergence in L2 of the sphere — which is to say that

The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:

If the coefficients decay in ℓ sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to ƒ.

A real square-integrable function ƒ can be expanded in terms of the real harmonics Y_ℓm above as a sum

Convergence of the series holds again in the same sense.