I'm not that familiar with Chebyshev polynomials, so I hope I'm not too far off.
Suppose that I have three order pairs $(x_0, f(x_0))$, $(x_1, f(x_1))$, and $(x_2, f(x_2))$ where $f : \mathbb{R} \to \mathbb{R}$. Letting $T_k$ denote the $k$-th Chebyshev polynomial, I can find the interpolating Chebyshev polynomial by solving the matrix equation \begin{equation} \left[ \begin{array}{ccc} T_0(x_0) & T_1(x_0) & T_2(x_0) \\ T_0(x_1) & T_1(x_1) & T_2(x_1) \\ T_0(x_2) & T_1(x_2) & T_2(x_2) \end{array} \right] \left[ \begin{array}{c} c_0 \\ c_1 \\ c_2 \end{array} \right] = \left[ \begin{array}{c} f(x_0) \\ f(x_1) \\ f(x_2) \end{array} \right] \end{equation}
I have two questions.
First, how can I find the coordinates in the monomial basis? I think this should be conceptually the same as change of basis in Euclidian space even though here we have spaces of functions, but I'm not 100% sure how to go about this.
Second, a paper that I am reading asks for the Chebyshev polynomials of $\log(x)$ - it says that it determines the coefficients for \begin{equation} f(x) = \sum_k \pi_k T_k(log(x)). \end{equation} Can this be achieved by replacing $x_i$ with $\log(x_i)$ in the matrix above?