Legendre series expansion
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When a problem lives on a sphere — the temperature on a planet, the gravitational field of a slightly squashed Earth, the radiation pattern of an antenna — the natural building blocks are not sines but Legendre polynomials. A Legendre series expansion writes a function on the interval [-1, 1] (typically the cosine of a polar angle) as a sum over those polynomials, the same way an ordinary Fourier series writes a periodic function as a sum of sines and cosines.
Legendre polynomials P_0, P_1, P_2, ... are the eigenfunctions of the Legendre Sturm-Liouville problem on [-1, 1] with weight w = 1, and they are orthogonal: integral from -1 to 1 of P_m P_n dx is zero for m not equal to n and equals 2/(2n+1) when m = n. So any square-integrable f on [-1, 1] expands as f(x) = sum over n of c_n P_n(x) with c_n = ((2n+1)/2) times integral from -1 to 1 of f(x) P_n(x) dx — a single projection per term, exactly as for sines, only with the Legendre normalization. Because the weight is 1, no extra factor appears in the integral, which makes Legendre series the most Fourier-like of the special-function expansions.
Its home is separation of variables in spherical coordinates. Solving Laplace's equation outside a charged or gravitating body, you substitute x = cos(theta) and the angular part becomes exactly the Legendre problem; the multipole expansion (monopole, dipole, quadrupole, ...) of a potential is a Legendre series in disguise. Convergence behaves like the classical Fourier case: smooth f converges fast and uniformly, but a jump in f produces Gibbs overshoot in the Legendre partial sums just as it does for sines.
Expand f(x) = |x| on [-1, 1]. By symmetry only even Legendre polynomials appear: |x| = (1/2) P_0 + (5/8) P_2 - (3/16) P_4 + ... The coefficients shrink, and partial sums approximate the V-shape, with a mild kink at x = 0 where |x| is not differentiable.
An even function expands in even Legendre polynomials only — parity halves the work, just as it does for Fourier cosine series.
A Legendre series is tailored to [-1, 1] with weight 1; do not confuse it with a Chebyshev series on the same interval, which uses the weight 1/sqrt(1 - x^2) and clusters its accuracy near the endpoints instead.