Homomorphisms from Functional Equations: The Goldie Equation, II

This first of three sequels to Homomorphisms from Functional equations: The Goldie equation (Ostaszewski in Aequationes Math 90:427–448, 2016) by the second author—the second of the resulting quartet—starts from the Goldie functional equation arising in the general regular variation of our joint paper (Bingham et al. in J Math Anal Appl 483:123610, 2020). We extend the work there in two directions. First, we algebraicize the theory, by systematic use of certain groups—the Popa groups arising in earlier work by Popa, and their relatives the Javor groups . Secondly, we extend from the original context on the real line to multi-dimensional (or infinite-dimensional) settings.