Interpolatory estimates for convex piecewise polynomial approximation

K. A. Kopotun*, D. Leviatan, I. A. Shevchuk

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, among other things, we show that, given r∈N, there is a constant c=c(r) such that if f∈C r [−1,1] is convex, then there is a number N=N(f,r), depending on f and r, such that for n≥N, there are convex piecewise polynomials S of order r+2 with knots at the nth Chebyshev partition, satisfying |f(x)−S(x)|≤c(r)(min⁡{1−x 2 ,n −1 1−x 2 }) r ω 2 (f (r) ,n −1 1−x 2 ), for all x∈[−1,1]. Moreover, N cannot be made independent of f.

Original languageEnglish
Pages (from-to)467-479
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume474
Issue number1
DOIs
StatePublished - 1 Jun 2019

Keywords

  • Convex approximation by polynomials
  • Degree of approximation
  • Jackson-type interpolatory estimates

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