4.5 Eigenfunction Expansions
The remaining property is the one that makes all of this useful: the eigenfunctions do not merely fail to overlap, they are enough to build any reasonable function.
Theorem 4.5.1 (Completeness). For a regular Sturm–Liouville problem the eigenvalues form an increasing sequence \(\lambda _1<\lambda _2<\cdots \to \infty \), and the corresponding eigenfunctions \(\{y_n\}\) form a complete orthogonal system in the space of functions square-integrable with respect to \(w\) on \([a,b]\). Every such \(f\) has the expansion \[f(x) = \sum ^{\infty }_{n=1}c_n\,y_n(x), \qquad c_n = \frac {\displaystyle \int ^{b}_{a}f(x)\,y_n(x)\,w(x)\,dx} {\displaystyle \int ^{b}_{a}y_n^{2}(x)\,w(x)\,dx},\] converging in the mean-square sense.
The formula for \(c_n\) should look familiar: it is the coefficient formula met for Legendre expansions in Section 3.2.3, for Hermite expansions in Section 3.6, and it is the Euler formula for Fourier coefficients in Chapter 5. All three are the same computation — project onto an orthogonal direction, divide by that direction’s squared length — carried out in different weighted spaces.
Remark 4.5.2. Completeness is a genuinely deeper statement than orthogonality and its proof is beyond these notes. Orthogonality says the eigenfunctions do not duplicate one another; completeness says nothing has been left out, that no non-zero function is orthogonal to every \(y_n\). It is completeness that entitles us to write an equality in Theorem 4.5.1 rather than merely an approximation, and it is what makes the method of separation of variables work: a solution expanded in eigenfunctions is not an approximation to the solution, it is the solution.
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