Journal of Inequalities and Applications
Volume 1 (1997), Issue 3, Pages 253-274
doi:10.1155/S1025583497000179
𝒪-regularly varying functions in approximation theory
Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, Kiel D-24098, Germany
Received 10 June 1996
Copyright © 1997 Stefan Jansche. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
For 𝒪-regularly varying functions a growth relation is introduced and characterized which gives an easy tool in the comparison of the rate of growth of two such functions at the limit
point. In particular, methods based on this relation provide necessary and sufficient conditions
in establishing chains of inequalities between functions and their geometric, harmonic, and
integral means, in both directions. For periodic functions, for example, it is shown how this
growth relation can be used in approximation theory in order to establish equivalence theorems
between the best approximation and moduli of smoothness from prescribed inequalities of
Jackson and Bernstein type.