Homomorphisms from functional equations: the Goldie equation

Indagationes Mathematicae

2018

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.S is continuous at 0 iff S(z n ) → 0, for some sequence z n ↑ 0, and then S is continuous everywhere, if finite-valued.The last part above draws on [HilP, Th. 2.5.2] that, for a subadditive function, continuity at the origin implies continuity everywhere. Theorem 0 above, in the presence of right-sided continuity, asserts that the merest hint of left-sided continuity gives full continuity; contrast this with the behaviour of the subadditive function 1 [0,∞) , which is continuous on the right but not on the left. This leads to the question of whether right-sided continuity can be thinned out. We are able to do so in the next two results below, but at the cost of imposing more structure, either on the left, or on the right. We need the following two definitions.Definitions. 1. Say that Σ is locally Steinhaus-Weil (SW), or has the SW property locally, if for x, y ∈ Σ and, for all δ > 0 sufficiently small, the sets Σ δ z := Σ ∩ B δ (z),

Section: Quantifier Weakening In Regular Variationmentioning

confidence: 99%

Additivity, subadditivity and linearity: Automatic continuity and quantifier weakening

Indagationes Mathematicae

2018

“…The special case (but nevertheless typical -see below) of h(t) = η 1 (t) ≡ 1 + t yields the circle product in a ring, a•b := a+b+ab -see [Ost4] for background. We recall also, from Javor [Jav] (in the broader context of h : E → F, with E a vector space over a commutative field F), that • h is associative iff h satisfies the Gołąb-Schinzel equation, briefly h ∈ GS (cf.…”

Section: Popa (Circle) Groupsmentioning

confidence: 99%

“…3), for which the auxiliary ψ(x) is necessarily Karamata regularly varying, so just as before (trivially, since ϕ ∈ SE) has exponential limit function, g ≡ e γ· say, and then (GBE-P ) simplifies to the original Goldie functional equation (see e.g. [BinO11,12], [Ost4]):…”

Section: Introductionmentioning

confidence: 98%

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Beurling moving averages and approximate homomorphisms

Indagationes Mathematicae

2016

Abstract. The theory of regular variation, in its Karamata and Bojanić-Karamata/de Haan forms, is long established and makes essential use of homomorphisms. Both forms are subsumed within the recent theory of Beurling regular variation, developed further here, especially certain moving averages occurring there. Extensive use of group structures leads to an algebraicization not previously encountered here, and to the approximate homomorphisms of the title. Dichotomy results are obtained: things are either very nice or very nasty. Quanti…er weakening is extended, and the degradation resulting from working with limsup and liminf, rather than assuming limits exist, is studied.

“…where f here is the function of primary interest ('G for Goldie, G for general': see e.g. [BinO6,10,11], [Ost2]). Specialising to ϕ ≡ 1, h ≡ 1 gives…”

Section: Introductionmentioning

confidence: 99%

General regular variation, Popa groups and quantifier weakening

Journal of Mathematical Analysis and Applications

2020

man and mathematician, who died with his boots on 'The soldiers' music and the rites of war Speak loudly for him.' (Shakespeare, Hamlet Act V.2) 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.