Differentiation using Chebyshev polynomials

Question

I have task - find derivative (degree N) using Chebyshev polynomials.

My issue is I can't find related theory. Give me a hint (book name, article, example), please.

I'm very appreciative.

Answer 1

Do you mean something like this? Express your function in a Chebyshev series. Differentiate term-by-term. If the coefficients decay rapidly enough, this is justified.

Basically the theory comes from Fourier series: the Chebyshev series of $f(x)$ on $[-1,1]$ corresponds to the Fourier series of $f(\cos(x))$ on $[0,\pi]$