caa a22 4500 605508267 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20150601xx s 000 0 eng 10.1007/s00010-014-0260-z doi (NATIONALLICENCE)springer-10.1007/s00010-014-0260-z Ostaszewski A. Mathematics Department, London School of Economics, Houghton Street, WC2A 2AE, London, UK aut Beurling regular variation, Bloom dichotomy, and the Gołąb-Schinzel functional equation [Elektronische Daten] [A. Ostaszewski] 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: $$\lambda (x)\lambda (y) = \lambda (x + y\lambda (x)),$$ λ ( x ) λ ( y ) = λ ( x + y λ ( x ) ) , known as the Gołąb-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 Gołąb, 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. Springer Basel, 2014 Beurling regular variation nationallicence Beurling's equation nationallicence self-neglecting functions nationallicence uniform convergence theorem nationallicence category-measure duality nationallicence Bloom dichotomy nationallicence Gołąb-Schinzel functional equation nationallicence Aequationes mathematicae Springer Basel 89/3(2015-06-01), 725-744 0001-9054 89:3<725 2015 89 10 https://doi.org/10.1007/s00010-014-0260-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0260-z text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Ostaszewski A. Mathematics Department, London School of Economics, Houghton Street, WC2A 2AE, London, UK aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/3(2015-06-01), 725-744 0001-9054 89:3<725 2015 89 10