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

Ostaszewski, A. J. (2015) Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation. Aequationes Mathematicae, 89 (3). pp. 725-744. ISSN 0001-9054

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Identification Number: 10.1007/s00010-014-0260-z


The class of 'self-neglecting' 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 'self-neglect' condition:(Formula presented.)known as the Goła{ogonek}b-Schinzel 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 one-parameter subgroups, is here accessory to the λ -Uniform Convergence Theorem (λ-UCT) for the recent, flow-motivated, '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 self-neglecting functions. The λ-UCT 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 self-neglecting functions.

Item Type: Article
Official URL:
Additional Information: © 2014 Springer Basel
Divisions: Mathematics
Subjects: G Geography. Anthropology. Recreation > GA Mathematical geography. Cartography
Date Deposited: 24 Jun 2014 14:30
Last Modified: 20 Oct 2021 02:14

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