I'm interested in finding a basis function $\phi(x)$, which I can use to approximate some function
$y(x) \approx \hat{y}(x) = \sum\limits_i c_i \phi_i((x - d_i)/s_i)$,
where its inverse function, $\phi^{-1}(x)$, is in the same basis. I.e.:
$\phi^{-1}(x) = l\phi((x - m)/n)$.
Does such a function exist? Or is there some way to prove it doesn't exist?
Consider $$\phi(x)=\begin{cases}-2x,\quad &x\le 0 \\ -x/2, \quad & x\ge 0\end{cases}$$ Clearly $\phi^{-1}=\phi$. And you can use the translations and dilations of $\phi$ to produce any piecewise linear function, up to an additive constant: indeed, $\phi'$ is a point mass at zero, which means that a linear combination of scaled and translated copies of $\phi$ can be any finite combination of point masses.
