### Functional Equations I: Cauchy’s Equation

#### Posted by Tom Leinster

This semester, I’m teaching a seminar course on functional equations. Why? Among other reasons:

Because I’m interested in measures of biological diversity. Dozens (or even hundreds?) of diversity measures have been proposed, but it would be a big step forward to have theorems of the form: “If you want your measure to have

*this*property,*this*property, and*this*property, then it must be*that*measure. No other will do.”Because teaching a course on functional equations will force me to learn about functional equations.

Because it touches on lots of mathematically interesting topics, such as entropy of various kinds and the theory of large deviations.

Today was a warm-up, focusing on Cauchy’s functional equation: which functions $f: \mathbb{R} \to \mathbb{R}$ satisfy

$f(x + y) = f(x) + f(y) \,\,\,\, \forall x, y \in \mathbb{R}?$

(I wrote about this equation before when I discovered that one of the main references is in Esperanto.) Later classes will look at entropy, means, norms, diversity measures, and a newish probabilistic method for solving functional equations.

Read on for today’s notes and an outline of the whole course.

I don’t want to commit to TeXing up notes every week, as any such commitment would suck joy out of something I’m really doing for intellectual fulfilment (also known as “fun”). However, I seem to have done it this week. Here they are. For those who came to the class, the parts in black ink are pretty much exactly what I wrote on the board.

Here’s the overall plan. We’ll take it at whatever pace feels natural, so the section numbers below don’t correspond to weeks. The later sections are pretty tentative — plans might change!

**Warm-up**Which functions $f$ satisfy $f(x + y) = f(x) + f(y)$? Which functions of two variables can be separated as a product of functions of one variable?**Shannon entropy**Basic ideas. Characterizations of entropy by Shannon, Faddeev, Rényi, etc. Relative entropy.**Deformed entropies**Rényi and “Tsallis” entropies. Characterizations of them. Relative Rényi entropy.**Probabilistic methods**Cramér’s large deviation theorem. Characterization of $p$-norms and power means.**Diversity of a single community**Background and introduction. Properties of diversity measures. Value. Towards a uniqueness theorem.**Diversity of a metacommunity**Background: diversity within and between subcommunities; beta-diversity in ecology. Link back to relative entropy. Properties.

## Re: Functional Equations I: Cauchy’s Equation

Fun old Putnam problem: Cauchy +

graph not dense$\implies$ $\mathbb{R}$-linear.