Where have all the functional equations gone (part III)

The last post ended with the following problem:

Problem: Find all continuous solutions to the following functional equation:

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

In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form $f(x) = cx$, but now we remove the assumption that the solution be continuously differentiable and ask whether the same conclusion holds. I found this problem to be extremely interesting, and at this point I will only give away that I eventually solved it, but after five (!) years.

In principle, it is plausible that, when one enlarges the space of functions in which one is searching for a solution from $C^1[-1,1]$ to the much larger $C[-1,1]$, then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps $\delta_1(t) = \frac{t+1}{2}$ and $\delta_2(t) = \frac{t-1}{2}$ on the space $[-1,1]$) is minimal, and therefore one expects the functional equation to be rigid enough to allow only for the trivial solutions (at least under some mild regularity assumptions). In short, a good case can be made in favor of either a conjecture that all the continuous solutions are linear or a conjecture that there might be new, nonlinear solutions.

1. Where did this problem come from

As I explained last time, Paneah studied these functional equations, and their generalizations in more complicated situation, because they arose from a problem in PDEs. The equations of interest were of the form

(CT) $f(t) = f\left(\delta_1(t)\right) + f\left(\delta_2(t)\right) \,\, , \,\, t \in [-1,1]$,

where $\delta_1, \delta_2$ satisfy certain conditions. This type of equation is called a Cauchy type functional equation, and the guided dynamical system which $\delta_1$ and $\delta_2$ generate plays an important role in their analysis. In that theory, some uniqueness results were obtained on the set of solutions of the functional equations (CT), but only under an additional continuous differentiablity assumption. Although in connection with the PDE problems that the functional equations came from the differentiability assumption was rather natural, it was still interesting to know whether the purely functional equations uniqueness result depended on this assumption. In other words, Paneah wanted to know whether the results he obtained in functional equations were “sharp”.

2. The solution to the problem

Five years passed from when Paneah first posed this problem to me until I solved it. Not that I was working on it non-stop: in the meantime I finished my undergraduate studies, quit my part time job in the industry, started a master’s under Paneah’s supervision, took some courses, solved some other problems, returned to a part time job in the industry, wrote my master’s thesis and defended it, started a PhD in another field (operator algebras, under the supervision of Baruch Solel), quit my job again, did some work on my PhD thesis, published my first papers, and had a life.

But I did come back to the problem once in a while, trying something new. And then I let it go again. I asked people about it, in functional equations and out of it. I know at least three other serious mathematicians who spent some time thinking about it (I am sure that this last sentence sounds ridiculous to non-mathematicians, but a lot of mathematicians would understand that it was not meant to be a joke). It reached the point where my curiosity grew stronger than my ambition and I would have been happy even if someone else solved it and just told me the answer. At one point I put up the following add in my department (I am reconstructing from memory):

Open Problem

I have a challenging open problem

Contact me: Orr Shalit, [my then email]

An undergraduate actually contacted me regarding this, but he never really got excited, I think.

Then one day I was trying to do something completely naive in operator algebras. I won’t get into details, but at the time I was thinking about operator algebras that are built from dynamical systems, and I was trying to improve some results of Davidson and Katsoulis. I wanted to play around with some concrete examples of dynamical systems from which to build these operator algebras. Well, my favorite examples of dynamical systems were the dynamical system which $\delta_1$ and $\delta_2$ generate in the interval $[-1,1]$, where $\delta_1$ and $\delta_2$ come from a Cauchy type functional equation (CT). So I started playing with these, trying just to tell when two such dynamical systems are isomorphic, and then, out of the blue, I realized the answer. This was perhaps the most stunning “Eureka” moment in my life, what a moment!

So I solved the problem and wrote a paper about, to which I will put a link after I warn you that the title of the paper gives the solution away. Here is a link to the paper on Banach Journal of Mathematical Analysis, an open access journal (no APCs!) in which I was very happy to publish at the time, but in which I did not publish since I became worried about some idiot in some committee frowning at my list of publications (I hope no idiot was insulted by the last comment).

I am still very proud of that paper, I think it is beautiful and clever. If I did not know better I would say that nobody was ever so proud of a result that nobody else cared about. But I know better…

Still, that moment of revelation when the answer unfolded before my eyes stays with me to this day. I sometimes return to it, for inspiration.