Journal of Inequalities and Applications (Jan 1999)

Global smoothness preservation and the variation-diminishing property

  • Gavrea Ioan,
  • Kacsó Daniela P,
  • Zhou Ding-Xuan,
  • Gonska Heinz H,
  • Cottin Claudia

Journal volume & issue
Vol. 1999, no. 2
p. 126510


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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.