CDAM: Computational, Discrete and Applicable Mathematics@LSE |
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CDAM Research Report, LSE-CDAM-2008-21September 2008 |
Regular variation without limits
N. H. Bingham and A. J. Ostaszewski
Karamata theory ([BGT] 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 ([BGT] Ch. 2) explores functions f for which the upper limit f*(λ) := lim sup f(λx)/f(x), 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) ≤ K λ&rho 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.A PDF file (248 kB) with the full contents of this report can be downloaded by clicking here.
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CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K. |
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