Any periodic function can be built by simply adding up trigonometric functions with different wavenumbers. Why is that possible? Because the inner product of trigonometric functions with different wavenumbers is 0 — in other words, they are orthogonal to one another. You will watch the number of added waves grow one at a time, see the inner product really come out to 0, and see that a partial differential equation falls apart into separate problems, one for each wave. Last comes the story of how the Fourier transform turns differentiation into division (Thomas–Fermi screening).
Mochizuki Research Group, Tokyo University of Science | From the lecture notes “Fourier Series” (1)–(14). The meaning of an orthogonal basis, the fact that ak and bk are weights for each number of waves, how the Fourier transform turns differentiation into division, and the derivation of the Thomas–Fermi screening formula are all on the explanation page (in Japanese).