# Approximate a known function by a sum of real power function

approximationregression

Given a known function $$f(x):[a,b]\to \mathbb{R}$$.
(For example, $$f(x) = e^{-\ln^2 \left(3x-\frac{3}{2} \right)} \mathbb{I}_{\{\frac{1}{2} over the support $$[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
$$f(x) \approx \sum_{n=1}^N c_i x^{ r_i}\mathbb{I}_{\{a_i
where $$r_i, c_i,a_i,b_i \in \Bbb R$$ and $$a\leq a_i 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
$$\underset{x\in[a,b]}{\text{max}} \left|f(x) – \sum_{n=1}^N c_i x^{r_i} \mathbb{I}_{\{a_i
with $$\epsilon \in \Bbb R$$ is given

Do you have any idea or reference for this problem?

This is not exactly an answer but it is too long for a comment.

Function $$\large f(x)=e^{-\log ^2\left(3 x-\frac{3}{2}\right)}$$ shows a maximum at $$x_*=\frac 56$$.

Expanding as a Taylor series around $$x_*$$ is almost correct over a very limited range of $$x$$.

However, the $$[2n,2n]$$ PadÃ© approximants $$P_n$$ do a pretty good job except very close to $$\frac 12$$. To give an idea, consider the norm $$\Phi_n=\int_{\frac{11}{20}}^2 \Big(f(x)-P_n\Big)^2\,dx$$

$$\left( \begin{array}{cc} n & \Phi_n\\ 1 & 7.4602\times 10^{-3} \\ 2 & 2.08463\times 10^{-5} \\ 3 & 1.72393\times 10^{-8} \\ 4 & 3.63413\times 10^{-12} \\ \end{array} \right)$$

For example, using $$x=t+\frac 56$$, $$\large P_2= \frac{1+6 t+\frac{1591 }{266}t^2-\frac{2409 }{266}t^3+\frac{19527 }{5320} t^4}{ 1+6 t+\frac{3985 }{266}t^2+\frac{4773}{266}t^3+\frac{7791}{760}t^4}$$