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.



Power series in functions other than monomials
Suppose you want to expand the function $f$ on an interval $I$ into powers of another function $h$ on $I$, which is invertible and maps $I$ onto an interval $J$. Let $H:=h^{-1}$ and $g:=f\circ H$, so that $f=g\circ h$. Thus, you want to expand $g(u)$ into powers of $u-u\_0$ for some $u\_0$ in the interior of $J$. The problem is to estimate the remainder, which reduces to bounding the $n$th derivative of $g$. If $g$ is analytic in an open disc $D\subset\mathbb C$ centered at $u\_0$ and we can bound $g$ on $D$, then we can use the Cauchy integral formula 
$$
g^{(n)}(u)=\frac{n!}{2\pi i}\int\_\gamma\frac{g(z)\,dz}{(z-u)^{n+1}}
$$ 
for (say) a circle $\gamma\subset D$
to bound $g^{(n)}$ in a neighborhood of $u\_0$, as desired. 


