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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(θ) z := \cos(\theta)

When we write the Chebyshev polynomials T nT_n as functions of θ\theta:

T n(z)=cos(nθ) T_n(z) = \cos(n \theta)

then the following series are equivalent:

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

and

f(cos(θ))= n=0 a ncos(nθ) f(\cos(\theta)) = \sum_{n = 0}^{\infty} a_n cos(n \theta)

References