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

## Abstract

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 |
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Official URL: | http://www.elsevier.com/wps/find/journaldescriptio... |

Additional Information: | © 2010 Elsevier Inc |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Date Deposited: | 23 Jul 2010 10:46 |

URL: | http://eprints.lse.ac.uk/28710/ |

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