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Regular variation without limits

Bingham, N. H. and Ostaszewski, Adam (2010) Regular variation without limits. Journal of Mathematical Analysis and Applications, 370 (2). pp. 322-338. ISSN 0022-247X

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Identification Number: 10.1016/j.jmaa.2010.04.013


Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 1]) explores functions f for which the limit function g(λ):=f(λx)/f(x) exists (as x→∞) and for which g(λ)=λρ subject to mild regularity assumptions on f. Further Karamata theory (N.H. Bingham et al. (1987) [8, Ch. 2]) explores functions f for which the upper limit , as x→∞, remains bounded. Here the usual regularity assumptions invoke boundedness of f* on a Baire non-meagre/measurable non-null set, with f Baire/measurable, and the conclusions assert uniformity over compact λ-sets (implying upper bounds of the form f(λx)/f(x)Kλρ for all large λ, x). We give unifying combinatorial conditions which include the two classical cases, deriving them from a combinatorial semigroup theorem. We examine character degradation in the passage from f to f* (using some standard descriptive set theory) and thus identify natural classes in which the theory may be established.

Item Type: Article
Official URL:
Additional Information: © 2010 Elsevier Inc
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 23 Jul 2010 10:46
Last Modified: 20 Jul 2021 02:06

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