### 9.3 Legendre and Chebyshev polynomials

For finite intervals, Legendre and Chebyshev polynomials are the ones most typically used. In the Chebyshev case, the polynomials themselves, their roots and quadrature points in general can be computed in closed form. They also satisfy a minmax property and lend themselves to using a fast Fourier transform (FFT).

#### 9.3.1 Legendre

Legendre polynomials correspond to the trivial weight function and the choice in Eqs. (9.28, 9.29, 9.30). The eigenvalues are

and the first two polynomials
A variation (in that it leads to non-monic polynomials) of the three term recurrence formula (9.40) is

#### 9.3.2 Chebyshev

Chebyshev polynomials correspond to the choice , and in Eqs. (9.28, 9.29, 9.30). In particular, the eigenvalues are

A closed-form expression for Chebyshev polynomials (which lends itself to the use of FFT) is
At first sight it might appear confusing that, given the above definition through trigonometric functions, are actually polynomials in (of degree , in fact). To get an idea of why this is so, we can compute the first few:
Notice from the above expressions that . In fact, a slight variation of the three-term recurrence formula (9.40) for Chebyshev polynomials becomes
As in the Legendre case, this is a variation of Eqs. (9.38, 9.39, 9.40) in that the resulting polynomials are not monic: the leading coefficient is given by
That is, the polynomial is monic.

Both Eq. (9.47) and Eq. (9.50) lead to the normalization

From the explicit expression (9.47), it can be noticed that the roots of are

These points play an important role below in Gauss quadratures and collocation methods (Section 9.4).

#### 9.3.3 The minmax property of Chebyshev points

Chebyshev polynomials satisfy a rather remarkable property in the context of interpolation. In Section 8.1 we pointed out that the error in polynomial interpolation of a function at nodal points satisfies [cf. Eq. (8.5)]

where .

When doing global interpolation, that is, keeping the endpoints fixed and increasing , it is not true that the error converges to zero even if the function is . For example, for each , could remain bounded as a function of ,

but could grow with . A classical example of non-convergence of polynomial interpolation on uniform grids is the following:

Example 53. Runge phenomenon (see, for instance, [154]).
Consider the function

and its interpolating polynomial [cf. Eq. (8.1)] at equally-spaced points
Then
where .

The error (8.5) can be decomposed into two terms, one related to the behavior of the derivatives of , , and another related to the distribution of the nodal points, . We assume in what follows that and that . The analogue results for an arbitrary interval can easily be obtained by a shifting and rescaling of coordinates. It can then be shown that, for all choices of nodal points,

Furthermore, the nodes, which minimize this maximum (thus the minmax term), are the roots of the Chebyshev polynomial of order , for which the equality is achieved:
when are given by Eq. (9.53); see, for example, [138], for the proof.

In other words, using Chebyshev points, that is, the roots of the Chebyshev polynomials, as interpolating nodes, minimizes the maximum error associated with the nodal polynomial term. Notice that, in this case, the nodal polynomial is given by .