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Chebyshev polynomials (changes)

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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)


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