This is something I recently started wondering about. I've long been interested in the idea of problems of the form "given a sequence of real numbers $a_n$, under what cases is there some way to interpolate it to fractional values of the index $n$, that is 'natural' somehow and, when there is, what is it?"
Now of course this is not really a "hard" formal question because it has a sort of subjective character to it, but in some cases these sequences do admit some solutions that are widely considered to be "natural" interpolations and what I wonder about is if there's ways to generalize them, to ask if there's a broader family of interpolations that is suggested by those that will allow us to interpolate other sequences. For example,
- Exponentiation: $a_n = \underbrace{e \cdot e \cdot e \cdots e}_{\text{$n$ copies of $e$}}$ becomes $a(x) = e^x$.
- Factorials: $a_n = 1 \cdot 2 \cdot 3 \cdots (n-1) \cdot n$ becomes $a(x) = \Gamma(x + 1)$
- Simple sums: $a_n = 1 + 2 + 3 + 4 + \cdots + (n-1) + n$ (aka "termial"), can be analyzed to be $\frac{n(n+1)}{2}$, which gives trivial extension $a(x) = \frac{x(x+1)}{2}$.
For example, $e^{0.3}$ is approximately 1.349859. $0.5!$ is approximately 0.886227 and interestingly has the exact expression $\frac{\sqrt{\pi}}{2}$, thanks to a relation to the famous Gaussian integral. And $\sum_{n=1}^{5.25} n$ can be considered to equal 16.40625 exactly.
Which of course makes one wonder about this sequence:
- Tetration: $^n a = \underbrace{a^{a^{a^{\cdots a^a}}}}_\text{$n$ copies of $a$}$ becomes $^x a =\ ???$
Now there's been some range of theorizations already made as to this, not all by mathematical professionals, but many of them are either incomplete or else have the disadvantage of being computationally expensive and opaque to the usual forms of analysis that we see with the other kind of functions such as Taylor expansions, integral representations, and the like.
Of course, the initial sequences alone are not enough to characterize any of the interpolants: there's $\beth_2$ generic real functions that coincide with them at integer values, and even if we allow continuity, differentiability, and the like there are still $\beth_1$ such. That cannot be gotten down to a finite number even considering them as recurrences such as $(n+1)! = (n+1) \cdot n!$ and demanding this carry over to the reals - instead, we need at least one other, far more stringent condition. In exponentiation, the condition $e^{x + y} = e^x e^y$ is sufficient, together with continuity, to characterize the interpolation at least on real number values. For factorials, we need it continuous and log-convex, that is, that $\ln \Gamma(x)$ is a convex function of $x$, together with the generalized recursion $\Gamma(x+1) = x \cdot \Gamma(x)$. Not sure about the sum (maybe the same thing and with log-convex works there too), but what I'm really interested in is the tetration.
And this is where I started thinking about something else: intuitively, if we think of alternate solutions to the various interpolation problems discussed above, they tend to bear some relation to the above "principal" forms in terms of modification by a general 1-cyclic function, in the case where the recurrence is required to hold. For example, $e^x$ and $\Gamma(x)$ have "imposters" of the form $\theta(x) e^x$ and $\theta(x) \Gamma(x)$ for some 1-cyclic $\theta(x)$, while tetration has imposters $\mathrm{tet}(x + \theta(x))$ for any chosen principal $\mathrm{tet}$. The way this can be understood more generally is that such a $\theta$ offset can be thought of as producing "wiggles" or "wobbles" in the graph when its amplitude becomes sufficiently large, and the solutions that are "preferred" above look to be "steady" in some regard. Hence, I theorized the idea of coming up with a way to measure "wobbliness" generically.
And one of the things that came to mind was, based on some previous experience with physics-related applications and with computer graphics involving spline curves, is the idea of what is called the bending energy of a curve. In particular, if we have a suitably smooth curve $C$, we can define its bending energy via
$$\mathrm{BE}[C] := \int_C |\kappa|^2 ds$$
where the integral is over the length of the curve and parameterized by the arc length elapsed, $s$. So the hunch that I had was to consider minimization of the bending energy. And playing with schemes to approximate this numerically and applying perturbations, what I find is that it appears the curves for (1), (2), and (3) actually do indeed appear to be minimal-energy smooth interpolations of their respective sequences. My question is, is of course, first of all:
- Can this be proven?
and a close follow up,
- Can we use this fact to compute the function?
The trick here is that while this looks like the classic "minimize a functional subject to a constraint" you see in the "calculus of variations", I am having trouble expressing the constraint in those terms. You see, when we use the Euler-Lagrange equations, which is what you usually would want to do, only the starting point and ending point of the curve are given, i.e. problems of the form
"extremize the functional $J$ of a curve $C(t)$, $t \in [0, 1]$, subject to that $C(0) = P_0$ and $C(1) = P_1$"
but in this case, it looks something more like,
"extremize $J$ of $C(t)$, $t \in [0, \infty)$, subject to the infinite number of 'pegging' constraints $C(0) = P_0$, $C(1) = P_1$, $C(2) = P_2$, etc."
and what I am wondering is, is there something analogous to the Euler-Lagrange equations for these kind of "multi-pegged" optimization problems and/or is there a neat way to translate them to an E-L problem?
Apparently, the very general case where we are just given the points alone as an unordered set and a general plane curve, does not presumably have a simple description, because it becomes equivalent to, or even implies, the "travelling salesman problem", which is famously NP-complete computationally. But here, we are dealing with a rather more constrained problem: at the very least, assuming we don't use the recurrence even, we are dealing with points whose order is specified in sequence (equiv. to the "salesman" visiting the "cities" in a pre-set order, and just needs to know what intermediates will make for the shortest overall trip) and even more, the curve must describe the graph of a function, and the points are organized as though from such, which means it can't "bend back on itself" or do "loop-dee-loops". Are these two constraints sufficient to alleviate the NP-hardness and, even better, bring it to the describability threshold of a simple differential/integral equation or the like?