Before moving on to equations in higher dimensions, we pause to look again at Fourier series, with any eye toward seeing if there are other sets of functions which behave like the functions discussed in Section 3.7. It may seem strange that our approach will hinge upon looking again at ODE eigenvalue problems. However, you will remember that, in Chapters 1 and 3, when dealing with the separated eigenvalue problems

y′′ + λy = 0, 0 ≤ x ≤ L, y(0) = 0 or y′(0) = 0,

y(L) = 0 or y′(L) = 0,

we found that the eigenvalues and eigenfunctions had a number of important properties-we list these and, in parentheses, we compare them with similar eigenproperties of matrices.