Given a known function $f(x):[a,b]\to \mathbb{R}$. (For example, $f(x) = e^{-\ln^2 \left(3x-\frac{3}{2} \right)} \mathbb{I}_{\{\frac{1}{2}<x<2\}} $ over the support $[a,b] = \left[\frac{1}{2} ; 2 \right] $)
I want to approximate the function $f(x)$, over its support $[a,b]$, by a sum of piecewise power function as follows $$f(x) \approx \sum_{n=1}^N c_i x^{ r_i}\mathbb{I}_{\{a_i<x<b_i\}}$$ where $r_i, c_i,a_i,b_i \in \Bbb R$ and $a\leq a_i <b_i \leq b$ for $i=1,...,N$
The problem can be seen as:
Find the smallest $N$ such that there exists $(r_i,c_i,a_i,b_i)_{i=1,..,N}$ satisfying
$$\underset{x\in[a,b]}{\text{max}} \left|f(x) - \sum_{n=1}^N c_i x^{r_i} \mathbb{I}_{\{a_i<x<b_i\}} \right| < \epsilon \tag{1} $$
with $\epsilon \in \Bbb R$ is given
Do you have any idea or reference for this problem? Thank you in advance!