On 3-monotone approximation by piecewise polynomials

D. Leviatan, A. V. Prymak

Research output: Contribution to journalArticlepeer-review

Abstract

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation.

Original languageEnglish
Pages (from-to)147-172
Number of pages26
JournalJournal of Approximation Theory
Volume133
Issue number2
DOIs
StatePublished - Apr 2005

Keywords

  • 3-Monotone approximation by piecewise polynomials
  • Degree of approximation

Fingerprint

Dive into the research topics of 'On 3-monotone approximation by piecewise polynomials'. Together they form a unique fingerprint.

Cite this