Homomorphisms from Functional Equations: The Goldie Equation, II

TL;DR

This paper extends the Goldie equation to multi-dimensional and infinite-dimensional contexts, utilizing algebraic structures like Popa and Javor groups to deepen the understanding of regular variation.

Contribution

It introduces an algebraic framework for the Goldie equation in higher dimensions, expanding previous work to more complex settings and group structures.

Findings

Extended Goldie equation to multi-dimensional spaces

Utilized Popa and Javor groups for algebraic analysis

Provided new insights into regular variation in advanced settings

Abstract

In this sequel to arXiv1407.4089 by the second author, we extend to multi-dimensional (or infinite-dimensional) settings the Goldie equation arising in the general regular variation of `General regular variation, Popa groups and quantifier weakening', J. Math. Anal. Appl. 483 (2020) 123610, 31 pp. (arXiv1901.05996). 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.