I've been asked to find the coefficients $c_0, c_1, c_2, c_3$ for $P_4(x) = c_0 \, T_0(x) + c_1 \, T_1(x) + c_2 \, T_2(x) +c_3 \, T_3(x)$ given $f(x) = e^{-2x}$, where $T_n(x)$ are Chebyshev polynomials.
I've tried to calculate the coefficients, but result in much larger than expected error. Any tips are appreciated!
If you consider the norm $$\Phi_n=\int_{-1}^{+1}\Big[e^{-2x}-\sum_{k=0}^n c_k\,T_k(x)\Big]^2\,dx$$ which is quite simple.
Minimize $\Phi_n$ setting to zero its derivatives with respect to the $c_k$ to obtain these coefficients.
For $n=3$ the largest absolute error should be $0.23$ at $x=-1$.
You probably have some mistake in the calculations.