Homomorphisms from functional equations: the Goldie equation

Abstract

The theory of regular variation, in its Karamata and Bojanic–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 Goląb–Schinzel functional equation (GS) and Goldie’s equation (GBE) below, are prominent there. Here we unify their treatment by ‘algebraicization’: extensive use of group structures introduced by Popa and Javor in the 1960s turns all the various (known) solutions into homomorphisms, in fact identifying them ‘en passant’, and show that (GS) is present everywhere, even if in a thick disguise.