Chapter 4
Sturm–Liouville Theory

Chapter 3 introduced six families of orthogonal polynomials one after another. Each was defined by a Rodrigues formula, each was shown to satisfy recurrence relations, and each was proved orthogonal on its own interval with respect to its own weight function. The table of Section 3.1 collected the intervals and the weights and said that particular families arise “depending on the choice of the interval \([a,b]\) and the weight function \(w(x)\)”.

That statement is true but it invites the wrong picture, and this chapter exists to correct it. The weight is not chosen. Once the differential equation a family satisfies is written down, the interval and the weight are determined, and the orthogonality is then automatic. All six families — and, as Chapter 5 will show, the trigonometric system behind Fourier series as well — are instances of one theorem.

That theorem is the subject of Sturm–Liouville theory.