Generalized Fourier series

Summary

A generalized Fourier series expands a vector of an inner product space along an orthogonal family, with coefficients computed by inner products. The classical Fourier series is the case of the exponentials $e^{inx}$ in $L^2$ of the circle. Other cases are expansions in Legendre polynomials, Bessel functions or eigenfunctions of a Sturm–Liouville problem. The partial sums are the best approximations from the span of the first terms, and the series recovers the vector when the family is complete.