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Let $A$ be a finite (and small-ish) set of positive real numbers and 0. Let $B$ be a subset of $\mathbb N^0$, up to some (small-ish) bound.

I have a function $f(t)$, $A \rightarrow B$ that is strictly monotonically increasing. It is $f(0) = 0$ and $f$ shall cover all values of its codomain $B$.

In plain English: $f$ starts at (0,0) and always increases by one, but with varying $\Delta t$ in between the steps.

A simple continuous version of $f(t)$ is $c(t)$ which I can get by piecewise linear interpolation. However since my $\Delta t$ values are all different, the derivative $c'(t) = dc/dt$ will have discontinuities.

I am looking for the easiest possible function $g(t)$ (stretching and squeezing $t$) such that $\frac {dc(g(t))}{dt}$ is smooth, with $g(t)=t$ $\forall t \in A$.

In plain English: How can I turn my simple linear interpolation into something that has a continuous time derivative (but still perfectly interpolates $f$)?

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  • $\begingroup$ Can your question be boiled down to this: you have a sequence $(0,a_1,a_2,\dots,a_n)$, and you want a smooth monotonic function $h:[0,a_n]\to\{0,1,\dots,n\}$ such that $h(a_i)=i$ for all $i$? Why does it have to be of the form $h=c\circ g$? $\endgroup$
    – user3883
    Jul 1 '18 at 5:34
  • $\begingroup$ If it is sufficient for $h$ to be $C^1$ and not $C^\infty$, you could try Fritsch & Carlson's monotone piecewise cubic interpolation. $\endgroup$
    – user3883
    Jul 1 '18 at 5:35
  • $\begingroup$ @Rahul yes, sorry, it should be $f:A \rightarrow B$. I just fixed it. $\endgroup$
    – rsp1984
    Jul 1 '18 at 14:03
  • $\begingroup$ @Rahul The codomain of $c(t)$ is no longer $B$ but rather $B$ and all real numbers in between. And yes, I want a smooth monotonic function the way you described, but one that maps $[0,a_n] \rightarrow [0 .. n]$. $\endgroup$
    – rsp1984
    Jul 1 '18 at 14:10
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Look up monotone splines, e.g. Wikipedia Monotone_cubic_interpolation.
(Normal cubic splines, see e.g. Numerical Recipes pages 120-122, are simpler but, surprisingly, may be non-monotone for monotone data.)

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