[Paper Review] Uniformity and self-neglecting functions: II. Beurling Regular Variation and the class {\Gamma}

Ν. H. Bingham, A. J. Ostaszewski|arXiv (Cornell University)|Jul 19, 2013

Advanced Topology and Set Theory16 references3 citations

TL;DR

This paper introduces Beurling regular variation as a generalization of Karamata regular variation, establishing a Uniform Convergence Theorem (UCT) for measurable or Baire functions in the class Γ. It proves that such functions admit a representation involving a slowly varying component and an integral of a vanishing function, extending de Haan’s gamma class theory and unifying Karamata and Beurling frameworks through the affine group action and cocycle dynamics.

ABSTRACT

Beurling slow variation is generalized to Beurling regular variation. A Uniform Convergence Theorem, not previously known, is proved for those functions of this class that are measurable or have the Baire property. This permits their characterization and representation. This extends the gamma class of de Haan theory studied earlier.

Motivation & Objective

• . The paper aims to generalize Beurling slow variation to a broader class of regularly varying functions using topological dynamics and cocycle theory.
• It seeks to establish a Uniform Convergence Theorem (UCT) for Beurling-regularly varying functions under measurability or Baire property assumptions.
• The objective includes characterizing and representing functions in the Beurling regular variation class, particularly extending de Haan’s gamma class.
• The work aims to unify Karamata and Beurling theories by embedding them within a common framework based on the affine group and Haar measure.
• It investigates how uniformity in auxiliary functions like ϕ ∈ SN propagates to the behavior of regularly varying functions f.

Proposed method

• . The authors use the affine group action on R+ to model the interplay between additive and multiplicative structures in regular variation.
• They define Beurling regular variation via the asymptotic behavior f(x + tϕ(x))/f(x) → g(t), where ϕ is self-neglecting (ϕ ∈ SN), generalizing Karamata’s multiplicative regular variation.
• A key technique is the use of the shift lemma and cocycle dynamics to analyze the convergence of h(x) = log f(x) under perturbations by ϕ(x).
• The proof relies on the Kestelman-Borwein-Ditor theorem and Baire’s category theorem to establish uniform convergence in the presence of the Baire property or measurability.
• The representation is derived by decomposing log f(x) into a slowly varying part and an integral of a function vanishing at infinity, using the structure of the Haar measure du/u and dv.
• The authors use the Miller homotopy framework to show that uniformity in ϕ ∈ SN implies uniform convergence in f, and vice versa, under appropriate conditions.

Experimental results

Research questions

• RQ1 . How can Beurling regular variation be systematically extended beyond Beurling slow variation to include non-zero indices?
• RQ2 . What conditions ensure uniform convergence of f(x + tϕ(x))/f(x) to a limit function g(t) for measurable or Baire functions?
• RQ3 . How does the class Γ of Beurling-regularly varying functions relate to de Haan’s gamma class and Karamata’s theory?
• RQ4 . In what way does the self-neglecting property of ϕ ∈ SN ensure uniformity in the convergence of f(x + tϕ(x))/f(x)?
• RQ5 . Can a representation theorem be established for Beurling-regularly varying functions that generalizes the de Bruijn-Karamata representation?

Key findings

• . A Uniform Convergence Theorem (UCT) is established for Beurling-regularly varying functions in class Γ under measurability or the Baire property, ensuring local uniform convergence of f(x + tϕ(x))/f(x) to g(t).
• . The representation f(x) = d(x) exp(ρ ∫₁ˣ du/ϕ(u)) exp(∫₀ˣ e(v)dv) holds, where d(x) → d ∈ (0, ∞), e(v) → 0, and ϕ ∈ SN, generalizing the Karamata and de Haan frameworks.
• . For ϕ ∈ SN and measurable f, the function ˜h(x) = log f(x) − ρ ∫₁ˣ du/ϕ(u) is slowly varying in the Karamata sense, enabling decomposition via the de Bruijn-Karamata representation.
• . The class Γ includes functions that are not necessarily slowly varying but satisfy a generalized regular variation condition with respect to a self-neglecting ϕ, extending the scope of classical regular variation.
• . The theory incorporates Karamata regular variation as a limiting case when ϕ(x) = x, formally recovering the standard multiplicative Karamata theory.
• . The converse holds: if f(x + uϕ(x))/f(x) → e^{ρu} locally uniformly, then ϕ must be self-neglecting, showing that uniformity in f implies uniformity in ϕ.