A known function that maps the interval $[-1,1]$ onto itself that is used to modify a linear gridding into an exponential one is given by,
\begin{align*} f \colon [-1,1] &\to [-1,1]\\ x &\mapsto f(x) := \frac{\sinh (\kappa x)}{\sinh (\kappa)}, \end{align*} where $\kappa$ parametrizes the "strength" of the exponential mapping, in the sense that $\lim\limits_{k \to 0}f(x)=x$. When sampling evenly-spaced points $x_i$, the points $f(x_i)$ will be concentrated around $x=0$.
I am attempting to generalise this mapping to a function $\bar{f}$ in the following:
- The mapping should go from arbitrary $[a,b] \to [a,b]$ (where $\bar{f}(a)=a, \bar{f}(b)=b$) for $a<b$.
- applying $\bar{f}$ to evenly spaced points $x_i$ aggregate them around some arbitrary point $x=c$, where $a<c<b.$
- Ideally, there should be a symmetry around $x=c$. That is, $\bar{f}(c-x) = \bar{f}(c+x)$.
- Ideally there should still be a limit in the parameter(s) so that we recover $\bar{f}(x) = x$ in that limit.
I naively attempted to simply make $\bar{f}$ a polynomial in $f$, but this does not seem to work: In practice, 1) is easy to do and basically just involves applying, for example, some linear mapping to $x$ and $f(x)$. 2) is essentially a third condition we impose on top of 1). This could be accomodated by making, for example, $\bar{f}(x)$ be a quadratic function of $f(x)$ instead, but then I don't see how to realise 3) and 4). In parallel, 3) is yet another equation to be satisfied that seems to preclude $\bar{f}(x)$ to be a linear function of $f(x)$, which seemingly runs into a contradiction with 4) as well.
Is there maybe some other functional form for $\bar{f}$?