Journal of Inequalities and Applications
Volume 4 (1999), Issue 2, Pages 91-114

Global smoothness preservation and the variation-diminishing property

Claudia Cottin,1 Ioan Gavrea,2 Heinz H. Gonska,3 Daniela P. Kacsó,2 and Ding-Xuan Zhou4

1Department of Mathematics and Technology, University of Applied Sciences (FH), Bielefeld D-33511, Germany
2Department of Mathematics, Technical University, Cluj-Napoca R-3400, Romania
3Department of Mathematics, Gerhard Mercator University, Duisburg D-47048, Germany
4City University of Hongkong, 83 Tat Chee Avenue, Kowloon, Hong Kong

Received 30 August 1998; Revised 28 September 1998

Copyright © 1999 Claudia Cottin et al. 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.


In the center of our paper are two counterexamples showing the independence of the concepts of global smoothness preservation and variation diminution for sequences of approximation operators. Under certain additional assumptions it is shown that the variation-diminishing property is the stronger one. It is also demonstrated, however, that there are positive linear operators giving an optimal pointwise degree of approximation, and which preserve global smoothness, monotonicity and convexity, but are not variation-diminishing.