Expressing a polynomial in terms of Chebyshev series

Question

How do I express an nth order polynomial in terms of the Chebyshev terms of the first kind? In other words, how do I express f(x) = $a_0$+$a_1$x+...+$a_nx^n$ in terms of $b_0T_0+b_1T_1+...+b_nT_n$?

Answer 1

This is first the part of the procedure for Economization of Power Series, see e.g. Abramowitz/Stegun 22.20. In Press et al., Numerical Recipes in C, 2nd ed, Ch. 5.11, you can find the function pccheb which computes the Chebyshev coefficients $b_k$ from the $a_k$. The main idea is a double loop based on the relation

$$ x^k = \frac{1}{2^{k-1}}\left(T_k(x) + {k \choose 1}T_{k-2}(x) + {k \choose 2}T_{k-4}(x) \cdots \right) $$