a generalized Fourier series
The ordinary Fourier series writes a function as a sum of sines and cosines. But sines and cosines are just one particular family of eigenfunctions — the ones for the simplest Sturm-Liouville problem. A generalized Fourier series does the same thing using whatever eigenfunctions the problem at hand actually produces: Bessel functions on a disk, Legendre polynomials on a sphere, and so on.
Given a complete orthogonal family {u_n} of Sturm-Liouville eigenfunctions with weight w, any reasonable function f is written f(x) = sum over n of c_n u_n(x), where the coefficients come straight from weighted orthogonality: c_n = (integral of f u_n w dx) / (integral of u_n^2 w dx). Each c_n is the 'amount' of f pointing along the direction u_n — a projection, exactly like resolving a vector into components along perpendicular axes. The eigenfunctions are the custom-built coordinate system adapted to the geometry and boundary conditions of your problem.
This is why generalized Fourier series are the universal output of separation of variables. You solve the spatial eigenvalue problem, get the natural modes u_n, expand both the initial data and the solution in them, and the PDE decouples into one simple ODE per mode. The completeness half of the Sturm-Liouville theorem guarantees the series really does represent f (in mean square), so nothing is lost. Ordinary trigonometric Fourier series, Fourier-Bessel series, and Legendre series are all the same idea wearing different eigenfunctions.
On a disk of radius 1 with f = 0 on the rim, expand a temperature profile as a Fourier-Bessel series f(r) = sum of c_n J_0(alpha_n r), where alpha_n are the zeros of J_0. Each c_n = (integral from 0 to 1 of f(r) J_0(alpha_n r) r dr) / (integral from 0 to 1 of J_0(alpha_n r)^2 r dr) — the weight r baked in.
Same recipe as ordinary Fourier, but the building blocks and the weight are dictated by the geometry.
The series converges to f in the mean-square (L^2) sense, which does not by itself guarantee pointwise convergence at every point — at a jump it converges to the average of the two sides, and Gibbs overshoot appears just as with trigonometric series. Smoother f gives stronger (uniform) convergence.