Homomorphisms from Functional Equations: The Goldie Equation

TL;DR

This paper unifies various functional equations related to regular variation by algebraic methods, revealing the pervasive presence of the Go{ }ab-Schinzel equation as a homomorphism within a group-theoretic framework.

Contribution

It introduces an algebraic approach that transforms solutions of functional equations into homomorphisms, unifying their treatment and highlighting the widespread role of the Go{ }ab-Schinzel equation.

Findings

Functional equations are unified via group structures.

The Go{ }ab-Schinzel equation is shown to be fundamental.

Homomorphisms characterize solutions across different equations.

Abstract

The theory of regular variation, in its Karamata and Bojani\'c-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Go{\l}\k{a}b-Schinzel functional equation (GS), are prominent there. Here we unify their treatment by `algebraicization': extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various solutions into homomorphisms, and show that (GS) is present everywhere, even if in a thick disguise.