Where have all the functional equations gone (part II)

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) $f(t) = f\left(\frac{t+1}{2} \right) + f \left(\frac{t-1}{2} \right) \,\, , \,\, t \in [-1,1] .$

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

(*) $g(t) = \frac{1}{2}g\left(\frac{t+1}{2}\right) + \frac{1}{2}g\left(\frac{t-1}{2}\right) \,\, , \,\, t \in [-1,1] .$

where $g = f'$, so $g$ 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 $g$ is a continuous solution of (*) then $g = const.$

We will prove the claim below. Assuming the claim for the moment, deduce that $f' = c$ for some $c \in \mathbb{R}$, thus $f(x) = cx + b$. Plugging in the original functional equation (FE) we find that $b = 0$. Thus, all continuously differentiable solutions of (FE) are of the form $f(x) = cx$, for some $c \in \mathbb{R}$. 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

dynamical system is a topological space $X$ and a family of continuous maps $\delta_1, \ldots, \delta_N : X \rightarrow X$. The main problem in topological dynamics is: given a point $x \in X$, how does it move around $X$ under the influence of the maps $\delta_1, \ldots, \delta_N$

Definition 1: Given a dynamical system $(X, \delta_1, \ldots, \delta_N)$ and a point $x \in X$the orbit of $x$ is the set

$O(x) = \{x\} \cup \{\delta_{i_1} \circ \cdots \delta_{i_k} (x) : k \in \mathbb{N}, i_1, \ldots, i_k \in \{1, \ldots, N\}\}.$

Thus, the orbit of $x$ is the set of all points in $X$ which can be reached from $x$ by iterating the maps $\delta_1, \ldots, \delta_N$.

Definition 2: A dynamical system $(X, \delta_1, \ldots, \delta_N)$ is said to be minimal if for all $x \in X$, the orbit of $x$ is dense in $X$, i.e.,

$\overline{O(x)} = X.$

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 $([-1,1], \delta_1, \delta_2)$, where

$\delta_1(t) = \frac{t+1}{2}$  and  $\delta_2(t) = \frac{t-1}{2}$.

One can use induction on $k$ to show that given any points $t_0,t_1 \in [-1,1]$, there are $i_1\ldots, i_k \in \{1,2\}$ such that

$|\delta_{i_k} \circ \cdots \circ \delta_{i_1}(t_0) - t_1| \leq 2^{k-1}$.

Thus this dynamical system is minimal.

3. Proof of Claim 1

Proof of Claim 1: It will be useful to consider the dynamical system $([-1,1],\delta_1, \delta_2)$ of the above Example. Let $t_0 \in [-1,1]$ be a point where $g$ attains its maximum. Then the functional equation (*) can hold at $t = t_0$ only if $g$ attains its maximum at $\delta_1(t_0)$ and at $\delta_2(t_0)$ — this readily follows from the fact that the functional equation merely says that $g(t_0)$ is the mean of $g(\delta_1(t_0))$ and $g(\delta_2(t_0))$. Repeating the argument, we find that the maximum of $g$ is attained at the points $\delta_1\circ \delta_1 (t_0), \delta_1 \circ \delta_2(t_0), \delta_2 \circ \delta_1(t_0)$ and  $\delta_2 \circ \delta_2(t_0)$. By induction, the maximum of $g$ is attained at all points in the orbit of $t_0$. But the Example above, the orbit of $t_0$ is dense in $[-1,1]$, thus $g$ attains this maximum on a dense set. Since $g$ 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 $P$-configurations.

As before, let $t_0$ be a point where $g$ attains its maximum. Then $g$ must attain its maximum also at $\delta_1(t_0)$. It follows inductively that $g$ attains its maximum at the sequence of points $\{ \delta_1^{(n)}(t_0)\}_{n=1}^\infty$, where $\delta_1^{(n)}$ denotes $\delta_1$ composed with itself $n$ times. But clearly $\delta_1^{(n)}(t_0) \rightarrow 1$, thus (as $g$ is continuous) the maximum of $g$ is achieved at the point $1$. But the same argument works for the minimum of $g$, so the minimum of $g$ must also be achieved at $1$. This is possible only of $g$ 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

(**)  $g(t) = a_1(t) g\left( \frac{t+1}{2} \right) + a_2(t) g\left(\frac{t-1}{2} \right)$,

where $a_1,a_2$ are two non-negative functions satisfying $a_1(t) + a_2(t) = 1$. Then both of the above approaches fail to prove that the only solutions to (**) are constants, because if, say $a_1(t_0) = 0$ (where $g(t_0) = \max g$) then we cannot conclude that the maximum of $g$ is attained at $\delta_1(t_0)$, 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 $(X,\delta_1, \ldots, \delta_N)$ together with closed sets $\Lambda_1, \ldots, \Lambda_N \subset X$ such that “one may use $\delta_i$ only on points $x \notin \Lambda_i$“. 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 $f$ to the equation

(FE) $f(t) = f\left(\frac{t+1}{2}\right) + f\left( \frac{t-1}{2}\right) \,\, , \,\, t \in [-1,1]$

are of the form $f(x) = cx$.

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 $f(x) = cx$?