eigenfunction expansion

Writing a function as an infinite series in the eigenfunctions of a Sturm-Liouville problem, with each coefficient found the same way a Fourier coefficient is.

An eigenfunction expansion (or generalized Fourier series) of $f$ in the eigenfunctions $y_1,y_2,\ldots$ of a Sturm-Liouville problem is $f(x)\sim\sum_{n=1}^\infty c_ny_n(x)$ , with $c_n=\dfrac{\int_a^b\sigma(x)f(x)y_n(x)\,dx}{\int_a^b\sigma(x)y_n(x)^2\,dx}$ -- the same projection formula a Fourier coefficient uses, generalized from sines and cosines to any Sturm-Liouville eigenbasis.