an eigenfunction expansion

A complicated sound can be split into pure tones — a fundamental plus overtones — and rebuilt by adding those tones back at the right volumes. An eigenfunction expansion does the same for a function: it breaks a given function into the natural modes of a boundary value problem and reconstructs it as a weighted sum of those modes. Each eigenfunction is a pure tone; the expansion is the recipe for how loudly each one plays.

Concretely, if y_1, y_2, y_3, ... are the eigenfunctions of a Sturm-Liouville problem on [a, b] with weight w, the expansion of a function f is f(x) = c_1 y_1(x) + c_2 y_2(x) + c_3 y_3(x) + ... The coefficients fall out of orthogonality: multiply both sides by y_n(x) w(x), integrate over [a, b], and every term on the right vanishes except the n-th, leaving c_n = (integral of f y_n w dx) / (integral of y_n^2 w dx). The denominator is just the weighted squared length of y_n; if the eigenfunctions are normalised it is 1 and the formula is even simpler. So computing the expansion is a sequence of weighted integrals, one per mode.

This is the single most important computational tool in the whole subject, because it converts a hard problem into many easy ones. To solve a heat or wave equation, you expand the initial data in eigenfunctions, evolve each simple mode on its own (each just rises, decays, or oscillates), and add the results. The ordinary Fourier sine and cosine series are the special case where the eigenfunctions are sines or cosines; the general theory says the same trick works for Bessel, Legendre, and every other Sturm-Liouville family.