I’ll start off exactly where I stopped in the previous post: I will tell you my solution to the problem my PDEs lecturer (and later master’s thesis advisor) Paneah gave us:
Problem: Find all continuously differentiable solutions to the following functional equation:
(FE) $latex f(t) = fleft(frac{t+1}{2} right) + f left(frac{t-1}{2} right) ,, , ,, t in [-1,1] .$
Before writing a solution, let me say that I think it is a fun exercise for undergraduate students, and only calculus is required for solving it, so if you want to try it now is your chance.
1. Solution of Problem
Solution: Well, the assumption of continuously differentiability (which we were left to impose ourselves) begs us to differentiate the equation to get
(*) $latex g(t) = frac{1}{2}gleft(frac{t+1}{2}right) + frac{1}{2}gleft(frac{t-1}{2}right) ,, , ,, t in [-1,1] . $
where $latex g = f’$, so $latex g$ is continuous. This can be considered as a functional equation in its on right, and our goal is now to find all continuous solutions to (*).
Claim 1: If $latex g$ is a continuous solution of (*) then $latex g = const.$
We will prove the claim below. Assuming the claim for the moment, deduce that $latex f’ = c$ for some $latex c in mathbb{R}$, thus $latex f(x) = cx + b$. Plugging in the original functional equation (FE) we find that $latex b = 0$. Thus, all continuously differentiable solutions of (FE) are of the form $latex f(x) = cx$, for some $latex c in mathbb{R}$. That any function of this form satisfies (FE) is obvious. Thus it remains to prove Claim 1. At this point it is convenient to introduce some terminology.
2. Brief on dynamical systems
A dynamical system is a topological space $latex X$ and a family of continuous maps $latex delta_1, ldots, delta_N : X rightarrow X$. The main problem in topological dynamics is: given a point $latex x in X$, how does it move around $latex X$ under the influence of the maps $latex delta_1, ldots, delta_N$?
Definition 1: Given a dynamical system $latex (X, delta_1, ldots, delta_N)$ and a point $latex x in X$, the orbit of $latex x$ is the set
$latex O(x) = {x} cup {delta_{i_1} circ cdots delta_{i_k} (x) : k in mathbb{N}, i_1, ldots, i_k in {1, ldots, N}}.$
Thus, the orbit of $latex x$ is the set of all points in $latex X$ which can be reached from $latex x$ by iterating the maps $latex delta_1, ldots, delta_N$.
Definition 2: A dynamical system $latex (X, delta_1, ldots, delta_N)$ is said to be minimal if for all $latex x in X$, the orbit of $latex x$ is dense in $latex X$, i.e.,
$latex overline{O(x)} = X.$
Minimal dynamical systems turn out to be very useful for proving uniqueness of solutions to certain functional equations, as we shall see.
Example: Consider the dynamical system $latex ([-1,1], delta_1, delta_2)$, where
$latex delta_1(t) = frac{t+1}{2}$ and $latex delta_2(t) = frac{t-1}{2}$.
One can use induction on $latex k$ to show that given any points $latex t_0,t_1 in [-1,1]$, there are $latex i_1ldots, i_k in {1,2}$ such that
$latex |delta_{i_k} circ cdots circ delta_{i_1}(t_0) – t_1| leq 2^{k-1}$.
Thus this dynamical system is minimal.
3. Proof of Claim 1
Proof of Claim 1: It will be useful to consider the dynamical system $latex ([-1,1],delta_1, delta_2)$ of the above Example. Let $latex t_0 in [-1,1]$ be a point where $latex g$ attains its maximum. Then the functional equation (*) can hold at $latex t = t_0$ only if $latex g$ attains its maximum at $latex delta_1(t_0)$ and at $latex delta_2(t_0)$ — this readily follows from the fact that the functional equation merely says that $latex g(t_0)$ is the mean of $latex g(delta_1(t_0))$ and $latex g(delta_2(t_0))$. Repeating the argument, we find that the maximum of $latex g$ is attained at the points $latex delta_1circ delta_1 (t_0), delta_1 circ delta_2(t_0), delta_2 circ delta_1(t_0)$ and $latex delta_2 circ delta_2(t_0)$. By induction, the maximum of $latex g$ is attained at all points in the orbit of $latex t_0$. But the Example above, the orbit of $latex t_0$ is dense in $latex [-1,1]$, thus $latex g$ attains this maximum on a dense set. Since $latex g$ is continuous, it must therefore be a constant. That concludes the proof of the claim, and therefore also concludes the solution of the problem.
4. Alternative solution to Problem
Here is a somewhat different solution to the problem, which (I later learned) is a simplification of Paneah’s approach to uniqueness in $latex P$-configurations.
As before, let $latex t_0$ be a point where $latex g$ attains its maximum. Then $latex g$ must attain its maximum also at $latex delta_1(t_0)$. It follows inductively that $latex g$ attains its maximum at the sequence of points $latex { delta_1^{(n)}(t_0)}_{n=1}^infty$, where $latex delta_1^{(n)}$ denotes $latex delta_1$ composed with itself $latex n$ times. But clearly $latex delta_1^{(n)}(t_0) rightarrow 1$, thus (as $latex g$ is continuous) the maximum of $latex g$ is achieved at the point $latex 1$. But the same argument works for the minimum of $latex g$, so the minimum of $latex g$ must also be achieved at $latex 1$. This is possible only of $latex g$ is constant.
5. Complications and guided dynamical systems
Before moving forward, let me briefly indicate what is the trickier problem that Paneah treated in his papers on this. Suppose that the functional equation (*) is replaced with
(**) $latex g(t) = a_1(t) gleft( frac{t+1}{2} right) + a_2(t) gleft(frac{t-1}{2} right) $,
where $latex a_1,a_2$ are two non-negative functions satisfying $latex a_1(t) + a_2(t) = 1$. Then both of the above approaches fail to prove that the only solutions to (**) are constants, because if, say $latex a_1(t_0) = 0$ (where $latex g(t_0) = max g$) then we cannot conclude that the maximum of $latex g$ is attained at $latex delta_1(t_0)$, in fact it might not. In fact, there could be non-constant solutions to (**). To decide when this happens one has to consider guided dynamical systems (as I call them), which were introduced by Paneah for this purpose. In a nutshell, a guided dynamical system is a dynamical system $latex (X,delta_1, ldots, delta_N)$ together with closed sets $latex Lambda_1, ldots, Lambda_N subset X$ such that “one may use $latex delta_i$ only on points $latex x notin Lambda_i$”. In other words, a guided dynamical system whose evolution has some obstructions: in a sense the evolution is not generated by a semigroup, but rather by something more like a semigroupoid. As I’ve said, Paneah used these systems to study functional equations (and he was able to apply these systems to problems in integral geometry and PDEs), and I also developed the theory to a small extent in my master’s thesis. As far as I know this interesting notion has not been studied by others (though I have seen once a work in a similar spirit, see this paper). I am not going to talk about generalizations in this direction any further. I will go on, but in another direction.
6. Another problem
Let me stop with mathematics and return to my story telling. After I showed Paneah my solution to the Problem above I was very pleased. To recap, the answer to the problem is
All continuously differentiable solutions $latex f$ to the equation
(FE) $latex f(t) = fleft(frac{t+1}{2}right) + fleft( frac{t-1}{2}right) ,, , ,, t in [-1,1]$
are of the form $latex f(x) = cx$.
Then Paneah challenged me further, and asked: well, what about the continuous solutions?
Well, what about them? Could there be continuous solutions to the functional equation (FE) which are not of the form $latex f(x) = cx$?
2 thoughts on “Where have all the functional equations gone (part II)”