# Math-Linux.com

Knowledge base dedicated to Linux and applied mathematics.

Home > Mathematics > Interpolation > Chebychev polynomials

## Chebychev polynomials

Chebychev polynomials are a useful and important tool in the field of interpolation. Indeed, in order to minimize the error in Lagrange interpolation, the roots of Chebychev polynomials are definitely the best suited points of interpolation.

### 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:

 0 1 1 2 3 4 5

### 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:

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: