### 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:

Problem 1: Let $\{a_n\}_{n=0}^\infty$ be an arbitrary sequence of real numbers. Does there exist a function $f \in C^\infty$ such that $f^{(n)}(0) = a_n$?

As usual, $f^{(n)}(0)$ denotes the $n$th derivative of $f$ at $0$.

Problem 2: Consider the triangle in the plane with vertices $(0,0), (1,0), (0,1)$. Fix a continuous function of two variables $f$ defined on this triangle. For every point $(x,1-x)$ on the hypotenuse (with $0 < x< 1$ ) form the rectangle $R_x$ which has vertices $(0,0), (x,0), (x,1-x), (0,1-x)$. Suppose that for every $x \in (0,1)$, one has $\int \int_{R_x} f(x,y) dx dy = 0 .$

Does it follow that $f = 0$?

This is impossible to understand without a figure. I am incapable of drawing one, so I ask the readers to draw one for themselves.

Problem 3: Solve the functional equation $F(x+y+z) - F(x+y) - F(x+z) - F(y+z) + F(x) + F(y) + F(z) = F(0)$.

Problem 1 is the famous Borel problem, which was also given to Paneah when he was a student (and he solved it). Problem 2 is a problem in the area called integral geometry, and Paneah treated a more complicated version of this problem in a series of papers, see this paper for example. Problem 3 is a rather classical functional equation, and is related to another strand of Paneah’s interests at the time, that of overdeterminedness of functional equations, see this paper.

I definitely thought about all three, but managed to solve the third one (my technique was to reduce it to a sequence of difference equations on finer and finer grids – a technique which for a while later I tried to apply to every single equation I met). The reader might want to think about these problems – they are all worthwhile – but below there is a problem I like better. Now at the end of each of Paneah’s lectures was a time when students came up to him, asked questions and presented to him solutions to problems that he gave. I came to him one of these times and handed in my solution neatly, and on the following week I was delighted to hear that my proof was correct! Then Paneah presented to the few of us that remained after the lecture the following problem:

Problem: Find all functions $f : [-1,1] \rightarrow \mathbb{R}$ that  satisfy the functional equation $f(t) = f\left(\frac{t+1}{2}\right) + f\left(\frac{t-1}{2}\right) \,\, , \,\, t \in [-1,1] .$

One of the students asked immediately “what is assumed about $f$?” and Paneah answered something in the spirit of assume whatever you need (which is advice I follow to this day in matters of research). I went home and decided to assume that $f$ was continuously differentiable (after all, this was a course in PDEs!). This type of assumption usually makes functional equations much easier, but this equation is a functional equation in a single variable, thus being able to differentiate does not make it completely trivial.

Before I tell you how I like to solve this equation (I will only do this in the next post), I wish to tell you what Paneah told us immediately when showing us this equation: this fits in a bigger picture. This equation is no other than the classic Cauchy functional equation, when restricted to a one dimensional curve. Recall that Cauchy’s functional equation (on a bounded domain) is the equation $f(x+y) = f(x) + f(y)$

for $x,y$ in the region (say) $|x|+|y| \leq 1$It is well known that if $f : [-1,1] \rightarrow \mathbb{R}$ is assumed continuous then it solves this equation if and only if it is of the form $f(t) = c t$ for some $c \in \mathbb{R}$ (it is also well known that the assumption can be weakened to boundedness, measurability, or having a non-dense graph). I will repeat: it is a nice exercise for students in a first course in calculus to show that a continuous solution $f$, satisfying the equation for all points $(x,y)$ in the two dimensional region $|x| + |y| \leq 1$, must have the form $f(t) = ct$

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

is nothing but Cauchy’s functional equation, with the difference that the points $(x,y)$ for which the equation should hold are taken from the one dimensional curve $\Gamma = \left\{\left(\frac{t+1}{2},\frac{t-1}{2}\right) : t \in [-1,1]\right\} .$

This fits into Paneah’s study of conditional Cauchy equations, see this paper (this theme was also studied by many others in the functional equations community, see “Functional equations on restricted domains”, Marek Kuczma, Aequationes mathematicae, Vol. 18, 1978). In my next post I will write about my favorite way to solve this equation under the assumption of continuous differentiability.

Before ending, I will tell you a sad story which everyone can learn from. Paneah’s class was rather special, and though we studied some equations that everybody does (like the heat equation and Laplace’s equation) there were also problems that are usually not (if ever) treated in courses on PDEs, like Goursat’s problem or Paneah’s solution to the third order hyperbolic equation in the plane. Thus, the notes one takes in this course were extremely valuable, since no textbook could help you prepare for the final exam. Now, in Israel every course has two official final examinations, and students are allowed to go to both. I went to the first one, but students who missed the first one or failed it could go to the second exam. One of the students who was preparing for the second exam and did not have a set of notes of his own, asked me to borrow my notes. Of course you can borrow them, just bring them back.  Well, the lousy schmuck took the notes and that’s the last I saw of them. As I said, this was the last semester, and since I continued to do a master’s degree in the same department, but never saw him again, I assume that my notes were helpful!