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The Azimuth Project

Chebyshev polynomials

# 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

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

## References

Created on March 11, 2011 10:47:55
by

Tim van Beek
(0.0.0.0)