If one needs to describe (and maybe compress) functions or data on a sphere, spherical harmonics are really a thing.

An alternative would be to construct a new function (or matrix) that is not only periodic in azimuth, but also in elevation (i.e., extend elevation to a full circle -pi to +pi). Then, one can simply compute two independent Fourie r transforms: along azimuth and along elevation. [1] The same idea works on matrices using the Discrete Fourier transform (DFT/FFT). However, you then have to accept things like that your data points are all equal at the poles.

[1] https://en.wikipedia.org/wiki/Double_Fourier_sphere_method