Functional equations

A functional equation is an equation whose unknowns are functions. Often analytic conditions like continuity or differentiability are imposed and analytic methods are used to solve the equations, even when the equations are algebraic. The subject of functional equations is obscure but has a habit of cropping up in unexpected places.

Theory

General

  • Aczél, 1966: Lectures on functional equations and their applications
    • The standard reference
    • Aczél & Dhombres, 1989: Functional equations in several variables (doi)
  • Castillo, Iglesias, Ruiz-Cobo, 2005: Functional equations in applied sciences (pdf)
    • Updated version of: Castillo & Ruzi-Cobo, 1992: Functional equations and modeling in science and engineering
  • Stetkaer, 2013: Functional equations on groups (doi)

Associativity

  • Bell, 2011: Associative binary operations and the Pythagorean Theorem (doi, pdf)
    • Inspired by: Berrone, 2009: The associativity of the Pythagorean Law (doi)
    • Elegantly characterizes the \(p\)-norms as associative, homogeneous, monotonic functions (Theorem 2)

Applications

Bayesian probability

In the Cox-Jaynes theorem , the laws of probability theory are derived from “commonsense” qualitative postulates, expressed in terms of the associativity equation and other functional equations. The result is inspiring but the delicacy of the assumptions has caused controversy .

  • Jaynes, 2003: Probability theory: The logic of science, Ch. 2: The quantitative rules (doi)
  • Terenin & Draper, 2017: Cox’s theorem and the Jaynesian interpretation of probability (arxiv)

Copulas and t-norms

Functional equations like the associativity equation show up in the study of copulas and triangular norms.