### Definition

Let . The Chebychev polynomial of degree is the map defined as follows:

### Explicit computation of Chebychev Polynomials

Let and .

where

is the floor function and

is the real part.

The following table gives the first Chebychev polynomials:

### Recurrence relation between Chebychev polynomials

**Proposition.** , and for any number

**Proof.** Let

and

. By means of trigonometry formulae, we have the following two equalities:

Adding these two equalities, one obtains:

Therefore, for any

in

:

**Proposition.**The coefficient of the

-degree term of

is

.

**Proof.** We shall proceed by induction. The coefficient of is . Assume that the coefficient of the -degree term of is . Then, given the previous recurrence relation, one can see that the coefficient of the -degree term is twice that of (. Thus, .

### Orthogonality of Chebychev polynomials

Chebychev polynomials are pairwise -orthogonal; that is, they are orthogonal with regard to a weighted function defined by:

In particular:

**Proof.** To compute the previous integral, we use the following substitution:

We thus have:

The conclusion is therefore obvious.

### Roots of Chebychev polynomials

**Proposition.** Let . has *exactly* simple roots defined as follows:

**Proof.** Let

and

:

Since

has degree

, the

are precisely the roots of

. They are simple roots since for all

, we have

### Extrema of Chebychev polynomials

**Proposition.** Let . has exactly extrema defined by:

**Proof.** Let

and

. The first derivative of

is:

Therefore:

**Proposition.**
**Proof.**
### The maximum reached by Chebychev polynomials

A forthwith consequence of the previous propositions is that:

### Best choice of points of interpolation of Lagrange polynomial

We have seen that if and , then:

The aim is to determine the points of interpolation

such that

for all

-degree monic polynomials

.

Via an affine substitution, the equivalent problem is:

We shall prove that the points of interpolation that verify this property are precisely te roots of the Chebychev polynomial

.

**Theorem.**

where

**Proof.**
Let

be the monic Chebychev polynomial associated to

. The roots of

are those of

and are defined by:

We

*a fortiori* have:

Additionally:

Since

where

We thus have to show that:

We shall proceed by contradiction by assuming that:

Since

and

are monic polynomials:

Also

If

is an even number:

If is an odd number:

Finally:

This means that there are

sign changes for the map

, and consequently,

has

roots. But,

has degree

. Therefore:

Finally:

which contradicts our assumption. We finally conclude that: