Beurling slow and regular variation

Bingham, Ν. H.; Ostaszewski, Adam

doi:10.1112/tlms/tlu002

Cited by 23 publications

(38 citation statements)

References 45 publications

(38 reference statements)

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“…Let us recall a few definitions (see [1], Sections 2.11 & 3.10, [2], or [12]). We can derive straightforward properties (see [1], Section 3.10, and [10]).…”

Section: Definitions and Examplesmentioning

confidence: 99%

New Results on the Order of Functions at Infinity

Cadena¹,

Kratz²,

Omey³

2017

SSRN Journal

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Recently, new classes of positive and measurable functions, M(ρ) and M(±∞), have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., , 2017. Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is developed in this paper, studying new classes of functions of the type lim x→∞ log U (x)/H(x) = ρ < ∞ for a large class of normalizing functions H. It provides subclasses of M(0) and M(±∞).

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“…Let us recall a few definitions (see [1], Sections 2.11 & 3.10, [2], or [12]). We can derive straightforward properties (see [1], Section 3.10, and [10]).…”

Section: Definitions and Examplesmentioning

confidence: 99%

New Results on the Order of Functions at Infinity

Cadena¹,

Kratz²,

Omey³

2017

SSRN Journal

View full text Add to dashboard Cite

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“…locally uniformly in t. For η ≡ 1, these specialize to the self-neglecting functions of Beurling (BGT 2.3.1, [Kor,IV.11]; cf. [BinO4]). For ϕ ∈ SE the limit η = η ϕ is necessarily in GS [Ost2].…”

Section: Popa Circle Groupsmentioning

confidence: 99%

“…For an interpretation of τ f , inspired by Beck [Bec], as the occupation time measure (of [0, x]) of the continuous f -flow: dx/dt = f (x), see [BinO6] (and [BinO4]).…”

Section: Popa Circle Groupsmentioning

confidence: 99%

“…The nearest to our theme of homomorphy is the paper of Kahlig and Schwaiger [KahS], which studies a sequence of deformations taking the equation (GS) below in the limit to the classical Cauchy functional equation (CF E). For us the equations arise first in the classical Karamata theory of Regular Variation (briefly, RV -see [BinGT], henceforth BGT, the standard text, and [BinO3] for updates) and now in the recently developed theory of Beurling RV, as in [BinO4,6], which includes the Karamata theory. In the latter context, the first functional equation (in the unknowns K and ψ), the generalized 'Goldie-Beurling equation', has the form…”

Section: Introductionmentioning

confidence: 99%

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Homomorphisms from functional equations: the Goldie equation

Ostaszewski

2015

Aequat. Math.

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Abstract. The theory of regular variation, in its Karamata and Bojanić-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Go÷¾ ab-Schinzel functional equation (GS) and Goldie's equation (GBE) below, are prominent there. Here we unify their treatment by 'algebraicization': extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various (known) solutions into homomorphisms, in fact identifying them 'en passant', and show that (GS) is present everywhere, even if in a thick disguise.

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“…the Darboux property) satis…es SN ; this may be viewed as a Bloom dichotomy: a Beurling-slow function is either self-neglecting or pathological -see [BinO9] (or the more detailed [BinO7] and [BinO8], to which we refer below) or Section 5. Although (BSV ) includes via ' = 1 the Karamata additive slow version (i.e.…”

Section: Regular Variation Self-neglecting and Beurling Functionsmentioning

confidence: 99%

Beurling slow and regular variation

Cited by 23 publications

References 45 publications

New Results on the Order of Functions at Infinity

New Results on the Order of Functions at Infinity

Homomorphisms from functional equations: the Goldie equation

Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

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