Functional equations

Putting the funk in functions

February 6, 2017 — February 19, 2017

functional analysis
Hilbert space

Miscellaneous tricks with functions. The purer side of the functional wrangling which gets you, e.g. variational approximation.

1 Dan Piponi’s functional logarithms

Nice hack, Dan Piponi – Logarithms and exponentials of functions:

A popular question in mathematics is this: given a function \(f\), what is its “square root” \(g\) in the sense that \(g(g(x))=f(x)\). […] I want to approach the problem indirectly. When working with real numbers we can find square roots, say, by using \(\sqrt{x}=\exp\left(\frac{1}{2}\log x\right)\). I want to use an analogue of this for functions. So my goal is to make sense of the idea of the logarithm and exponential of a formal power series as composable functions.

2 Tom Leinster’s course

Tom Leinster, taught a punchy course on functional equations (course notes here):

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

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

He goes on to talk about Shannon entropy from a functional equation perspective, which is a refreshing derivation.

3 References

Aubrun, and Nechita. 2011. The Multiplicative Property Characterizes \(\ell_p\) and \(L_p\) Norms.” Confluentes Mathematici.
Granas, and Dugundji. 2003. Fixed Point Theory. Springer Monographs in Mathematics.