Some positive results and counterexamples in comonotone approximation, II

D. Leviatan, I. A. Shevchuk*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let f be a continuous function on [-1, 1], which changes its monotonicity finitely many times in the interval, say s times. In the first part of this paper we have discussed the validity of Jackson type estimates for the approximation of f by algebraic polynomials that are comonotone with it. We have proved the validity of a Jackson type estimate involving the Ditzian-Totik (first) modulus of continuity and a constant which depends only on s, and we have shown by counterexamples that in many cases the Jackson estimates involving the DT-moduli do not hold when there are certain relations between s, the number of changes of monotonicity, and r, the number of derivatives of the approximated function. Here we deal with all other cases and we obtain Jackson type estimates involving modified DT-moduli. We also provide counterexamples to complete the picture. Our technique for the positive results involves a two-tier approach. We first approximate the given function by comonotone piecewise polynomials which yield good approximation and then we replace the latter by polynomials.

Original languageEnglish
Pages (from-to)113-143
Number of pages31
JournalJournal of Approximation Theory
Volume100
Issue number1
DOIs
StatePublished - Sep 1999

Keywords

  • Comonotone approximation
  • Polynomial approximation

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