General Regular Variation, Popa Groups and Quantifier Weakening

TL;DR

This paper introduces a unified theory of regular variation using Popa groups, simplifying the understanding of quantifier weakening across various existing theories in the field.

Contribution

It develops a general framework of regular variation that encompasses Karamata, Bojanic-Karamata/de Haan, and Beurling theories through Popa groups and their harmonic analysis.

Findings

Unified approach simplifies regular variation theory

Encompasses multiple existing theories as special cases

Enhances understanding of quantifier weakening

Abstract

We introduce general regular variation, a theory of regular variation containing the existing Karamata, Bojanic-Karamata/de Haan and Beurling theories as special cases. The unifying theme is the Popa groups of our title viewed as locally compact abelian ordered topological groups, together with their Haar measure and Fourier theory. The power of this unified approach is shown by the simplification it brings to the whole area of quantifier weakening, so important in this field.