Thorough, accessible material about Fourier basis functions and Spherical Harmonics? Spherical harmonics appear in several computer graphics techniques.
I feel that in order to be a better computer graphics developer, I need to have a deep understanding of what they are and how they are used.

It seems that the reference most often recommended to understand Spherical Harmonics is "Stupid Spherical Harmonics Tricks" by Peter-Pike Sloan.

I started reading it but did not find a "satisfying" definition of SH, it seems like the document mostly relies on other references for the "basics".
Other references introduce the Fourier Basis functions as a "simpler version" of SH, but once again it seems hard to find good material explaining them.

What are thorough, accessible references to understand Fourier basis functions and Spherical Harmonics ?
wil, you largely have enough scholar background, you must have done Fourier and Laplace transforms in second year, and maybe in your engineering school again as part of signal processing classes.

If you read "stupid tricks" there is not much more you can do to find a condensed course at this point.

The second most famous paper that goes with SH for graphics is by Robin Green called "the gritty details":
http://www.research.scea.com/gdc2003/spherical-harmonic-lighting.pdf

And the third most important is the one by Ramamoorthi (the original paper preceding "An Efficient Representation for Irradiance Environment Maps"), which was called On the relationship between radiance and
irradiance: determining
the illumination from
images of a convex Lambertian object

And I think they mention somewhere that SH were previously most used by another science field, forgot which one, physics maybe, and that most of their base material came from these papers. So if you want to dig in the roots, you've got to pull out these mid last century references.