4.4 The Classical Families as Sturm–Liouville Problems

We can now do what the chapter promised: display the six families of Chapter 3, and the trigonometric system of Chapter 5, as one object.


Family	\([a,b]\)	\(p(x)\)	\(w(x)\)	\(\lambda _n\)

Legendre	\([-1,1]\)	\(1-x^{2}\)	\(1\)	\(n(n+1)\)
Jacobi	\([-1,1]\)	\((1-x)^{\alpha +1}(1+x)^{\beta +1}\)	\((1-x)^{\alpha }(1+x)^{\beta }\)	\(n(n+\alpha +\beta +1)\)
Chebyshev I	\([-1,1]\)	\(\left (1-x^{2}\right )^{1/2}\)	\(\left (1-x^{2}\right )^{-1/2}\)	\(n^{2}\)
Chebyshev II	\([-1,1]\)	\(\left (1-x^{2}\right )^{3/2}\)	\(\left (1-x^{2}\right )^{1/2}\)	\(n(n+2)\)
Hermite	\((-\infty ,\infty )\)	\(e^{-x^{2}/2}\)	\(e^{-x^{2}/2}\)	\(n\)
Laguerre	\([0,\infty )\)	\(x^{\alpha +1}e^{-x}\)	\(x^{\alpha }e^{-x}\)	\(n\)

Trigonometric	\([0,L]\)	\(1\)	\(1\)	\(\left (n\pi /L\right )^{2}\)

Table 4.1: Every orthogonal system in these notes as a Sturm–Liouville problem. The weight column reproduces the table of Section 3.1; it is now a consequence rather than a choice. The last row is the system underlying Fourier series.

Note 4.4.1. Read Table 4.1 alongside the table of Section 3.1 and the point of this chapter is visible at a glance. That table had two columns, an interval and a weight, and no reason for either. This one has the differential operator as well, and the weight is now computed from it by Theorem 4.2.1. The orthogonality that Chapter 3 established six times over is one application of Theorem 4.3.2.

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