### 9.2 Some properties of orthogonal polynomials

Given any weighted scalar product as in Eq. (9.17), the expansion of a function into polynomials is optimal in the associated norm if orthogonally projected to the space of polynomials of degree at most . More precisely, given a function on some interval and an orthonormal basis of polynomials , where has degree , its orthogonal projection,
onto the subspace spanned by satisfies
or
for all in the span of , that is, it minimizes the error for such .

The operator (9.33) is a projection in the sense that

and it is orthogonal with respect to : the residual satisfies
Notice that, unlike the interpolation problem (discussed below) here, the solution of the above least-squares problem (9.34) is not required to agree with at any prescribed set of points.

In order to obtain an orthonormal basis of , a Gram–Schmidt procedure could be applied to the standard basis . However, exploiting properties of polynomials, a more efficient approach can be used, where the first two polynomials are constructed and then a three-term recurrence formula is used.

In the following construction, each orthonormal polynomial is chosen to be monic, meaning that its leading coefficient is one.

• The zero-th-order polynomial:

The conditions that has degree zero and that it is monic only leaves the choice

• The first-order one:

Writing the condition yields

• The higher-order polynomials:

Theorem 19 (Three-term recurrence formula for orthogonal polynomials). For monic polynomials , which are orthogonal with respect to the scalar product , where each is of degree , the following relation holds

for .

Proof. Let . Since is a polynomial of degree , it can be expanded as

where the orthogonality of the polynomials implies that for . However, since and can be expanded in terms of the polynomials , it follows again by the orthogonality of that for . Finally, since and are both monic. This proves Eq. (9.40). □

Notice that , as defined in Eq. (9.40), remains monic and can therefore be automatically used for constructing , without any rescaling.

Eqs. (9.38, 9.39, 9.40) allow one to compute orthogonal polynomials for any weight function , without the expense of a Gram–Schmidt procedure. For specific weight cases, there are even more explicit recurrence formulae, such as those in Eqs. (9.43, 9.44) and (9.48, 9.49, 9.50) below for Legendre and Chebyshev polynomials, respectively.