Chebyshev series

Summary

A Chebyshev series expands a function on $[-1, 1]$ as a sum $\sum a_k T_k(x)$ of Chebyshev polynomials. Under the substitution $x = \cos \theta$ it becomes the Fourier cosine series of $f(\cos \theta)$. Truncated Chebyshev series are cheap to compute and are close to minimax approximations, within a factor that grows like $\log n$. So they are central in numerical analysis.