Noncommutative Analysis

Category: Paneah

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.

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Where have all the functional equations gone (part I)

My first encounter with research mathematics was in the last term of my undergraduate studies (spring 2003). My professor in the course “Introduction to Partial Differential Equations”, Prof. Boris Paneah, thought that it is pointless to give standard homework problems to students of pure mathematics, and instead he gave us several problems which were either extremely challenging, related to his research or related to advanced courses that he was going to give. This was a thrilling experience for me, and is one of the reasons why I decided not long after to do my master’s thesis under his supervision, since no other faculty member came even close to engaging us like Paneah (another reason was that the lectures themselves were fantastic). For example he suggested that we explore the ultrahyperbolic equation

u_{tt} + u_{ss} - u_{xx} - u_{yy} = 0 ,    in     \mathbb{R}^4,

or that we try to prove the existence of solutions to the two dimensional heat equation in a non-rectangular bounded region of the plane. I remember spending hours on the heat equation, unsuccessfully of course (if I was successful I would have probably become a PDE person). Especially memorable is the one time that he ended a lecture with the following three problems, which were, as you may guess, quite unrelated to the content of the lecture: Read the rest of this entry »