I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.

Consider the interval $[-\pi,\pi]$ let's say.

The Stone Weierstrass theorem tells us that the linear hull of functions

$$f_n(x)=(\arctan(x))^n$$ for $n \in \mathbb N_0$ is dense in the continuous functions on $[-\pi,\pi].$ So it makes sense to use these functions to approximate continuous functions on this interval.

On the other hand, we know that for the smaller class of $C^{\infty}$ functions on $(-\pi,\pi)$ Taylor's formula holds, i.e. for $f\in C^{\infty}$

$$ f(x) = \sum_{i=0}^k a_i x^i + M_k$$ and $M_k$ can be explicitly estimated using the mean value theorem.

Now I would like to understand whether one can get also a Taylor-type formula

$$f(x) = \sum_{i=0}^k b_i f_i(x) + N_k$$

such that $N_k$ can be explicitly estimated?

I figured out already a way to compute the coefficients $b_i$ in that case.

Namely, since $f_n$ has a Taylor expansion $$\arctan(x)^n=x^n(1-\frac{nx^2}{3}+\mathcal O(x^4))$$

you can recursively express the $a_i$ in terms of $b_i.$ The problem is that I do not know how to estimate the error $N_k$ this way.

Thus, I would like to understand whether one can get for $f \in C^{\infty}[-\pi,\pi]$ a nice explicit power series expansion in terms of functions $f_i$ with full control on the error.