# Contents

## Idea

Chebyshev polynomials are polynomials that are often used in approximations, for example in spectral methods.

An approximation with Chebyshev polynomials is essentially the same as an approximation with a Fourier series after a change of variable.

The mapping is:

$z := \cos(\theta)$

When we write the Chebyshev polynomials $T_n$ as functions of $\theta$:

$T_n(z) = \cos(n \theta)$

then the following series are equivalent:

$f(z) = \sum_{n = 0}^{\infty} a_n T_n(z)$

and

$f(\cos(\theta)) = \sum_{n = 0}^{\infty} a_n cos(n \theta)$