The nontrivial (nonzero) solutions , , of the SturmLiouville boundary value problem only exist at certain , . is called eigenvalue and is the eigenfunction.
The eigenvalues of a SturmLiouville boundary value problem are nonnegative real numbers. In addition, the associated eigenfunctions are orthogonal to each other with respect to the weighting function ,
The complete set of the solutions forms a complete orthogonal set of functions defined on the interval . Therefore, a piecewise continuous function can be expressed in terms of , , such that
where
The completeness allows us to express any piecewise continuous function in terms of these eigenfunctions while the orthogonality makes the expression unique and compact (no redundant terms). In addition, it can be shown that the orthogonal series is the best series available, i.e., each additional term fine tunes but not overhauls the sum of the existing terms. These properties generalize the conventional Fourier series and to any complete orthogonal series . The series is hence called the generalized Fourier series. The method of forming solutions by the general Fourier series is called the method of eigenfunction expansion which is an important technique in solving partial differential equations.
Examples of generalized Fourier series can be found in Bessel functions, Legendre polynomials, and other orthogonal polynomials such as Laguerre polynomials, Hermite polynomials, and Chebyshev polynomials. Each of these polynomials represents a complete orthogonal set in different coordinates or circumstances and can be considered as a special case of the SturmLiouville boundary value problem.
