Orthogonal polynomials on an interval

It was shown in Chapter 3 that there are three cases in which the eigenfunctions of a second-order ordinary differential operator that is symmetric with respect to a weight are polynomials. The polynomials in the three cases are the classical orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, and Jacobi polynomials. Each of these sets of polynomials is an example of a family of polynomials that are orthogonal with respect to an inner product that is induced by a positive weight function w on an interval of the real line. The basic theory of general orthogonal polynomials of this type is covered in this chapter. This includes expressions as determinants, three-term recurrence relations, properties of the zeros, and basic asymptotics. It is shown that under a certain condition on the weight w ( x ), which is satisfied in each of the three classical cases, each element of the space L w 2 can be expanded in a series using the orthogonal polynomials, analogous to the Fourier series expansion. These results carry over to more general measures than those of the form w ( x ) dx . Orthogonal polynomials occur naturally in connection with approximating the Stieltjes transform of the weight function or measure. This transform can also be viewed as a continued fraction. The central role played by the three-term recurrence relations leads to the question: do such relations characterize orthogonal polynomials? The (positive) answer is known as Favard's theorem. The chapter concludes with a brief discussion of the asymptotic distribution of zeros. Weight functions and orthogonality Let w ( x ) be a positive weight function on an open interval I =( a , b ) and assume that the moments are finite. Let Δ −1 = 1 and let Δ n , n ≥ 0, be the determinant The associated quadratic form is positive definite, so the determinant Δ n is positive. Consider the Hilbert space L w 2 , with inner product The polynomial is orthogonal to x m , m < n , while ( Q n , x n ) = Δ n . To see this, expand the determinant (4.1.2) along the last column. Computing the inner product of Q n with x m results in a determinant in which the last column of the determinant (4.1.1) has been replaced by column m +1 of the same determinant.