Ostaszewski, A. J.
(2015)
Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation.
Aequationes Mathematicae, 89 (3).
pp. 725744.
ISSN 00019054
Abstract
The class of 'selfneglecting' functions at the heart of Beurling slow variation is expanded by permitting a positive asymptotic limit function λ(t), in place of the usual limit 1, necessarily satisfying the following 'selfneglect' condition:(Formula presented.)known as the Goła{ogonek}bSchinzel functional equation, a relative of the Cauchy equation (which is itself also central to Karamata regular variation). This equation, due independently to Aczél and Goła{ogonek}b, occurring in the study of oneparameter subgroups, is here accessory to the λ Uniform Convergence Theorem (λUCT) for the recent, flowmotivated, 'Beurling regular variation'. Positive solutions, when continuous, are known to be λ(t) = 1 + at (below a new, 'flow', proof is given); a = 0 recovers the usual limit 1 for selfneglecting functions. The λUCT allows the inclusion of Karamata multiplicative regular variation in the Beurling theory of regular variation, with λ (t) = 1 + t being the relevant case here, and generalizes Bloom's theorem concerning selfneglecting functions.
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