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

General regular variation, Popa groups and quantifier weakening

“…Above we have used the Popa operation as a simplifying notational device for the regular variation above (general or otherwise), involving limits as x → ∞. But its usefulness is far greater, and is not confined to limits, as emerged in [BinO7], [Ost1]. Here one allows other auxiliary functions h, with corresponding circle operations • h .…”

Homomorphisms from functional equations: the Goldie equation

“…We turn to some background. The functions in GS have their origin for RV in the asymptotic analysis of self-equivarying functions ϕ, briefly ϕ ∈ SE [Ost2], which for some function η > 0 satisfy…”

“…[BinO4]). For ϕ ∈ SE the limit η = η ϕ is necessarily in GS [Ost2]. Only (CF E) visibly identifies its solution K as a homomorphism -of the additive group (R, +) -whereas homomorphy is a central feature in the recent topological development of the theory of regular variation [BinO1,2], [Ost1].…”

“…The second functional equation of concern replaces the K on the right of (GBE) by a further unknown function κ, yielding a natural 'Pexiderized' generalization 1 K(x + yη(x)) − K(y) = ψ(y)κ(x) (x, y ∈ R), (GBE-P ) considered also in [ChuT]. Passage to this more general format is motivated by a desire to include (GS), as the case K = ψ = η and κ = η − 1, which turns out to be highly thematic (see [Ost2], and Theorem 1 ′ ). Note that in this specialization the corresponding derivatives are identical: K ′ = κ ′ ; cf.…”

Beurling moving averages and approximate homomorphisms

“…and ϕ(x) = o(x). This traditional restriction may be usefully relaxed in two ways, as in [Ost3]: firstly, in imposing the weaker order condition ϕ(x) = O(x), and secondly by replacing the limit 1 by a general limit function η > 0, so that for A = [0, ∞)…”