The Goldie Equation: III. Homomorphisms from functional equations

Abstract

This is the second of three sequels to (Ostaszewski in Aequat Math 90:427–448, 2016)—the third of the resulting quartet—concerning the real-valued continuous solutions of the multivariate Goldie functional equation (GFE) below of Levi–Civita type. Following on from the preceding paper (Bingham and Ostaszewski in Homomorphisms from Functional Equations: II. The Goldie Equation, arXiv:1910.05816), in which these solutions are described explicitly, here we characterize (GFE) as expressing homomorphy (in all but some exceptional “improper” cases) between multivariate Popa groups, defined and characterized earlier in the sequence. The group operation involves a form of affine addition (with local scalar acceleration) similar to the circle operation of ring theory. We show the affine action in $$(GFE)\ $$ ( G F E ) may be replaced by a general continuous acceleration yielding a functional equation (GGE) which it emerges has the same solution structure as (GFE). The final member of the sequence (Bingham and Ostaszewski, The Gołąb–Schinzel and Goldie functional equations in Banach algebras, arXiv:2105.07794) considers the richer framework of a Banach algebra which allows vectorial acceleration, giving the closest possible similarity to the circle operation.