Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation

Ranko Bojanic, Fuhua Cheng

Research output: Contribution to journalArticlepeer-review

70 Scopus citations

Abstract

For any x ε{lunate} (0, 1) we first prove that if f{hook}x(t) ≡ |t - x| on [0, 1] then the Bernstein polynomials of f{hook}x satisfy the asymptotic relation ∑k = 0n | k n - x|(kn) xk(1 - x) n - k = (2x (1 - x) π) 1 2 1 √n + O( 1 n). This asymptotic relation is then used to study the rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. An estimate of the rate of convergence is given. This estimate is asymptotically the best possible at points where f{hook}′ is continuous.

Original languageEnglish
Pages (from-to)136-151
Number of pages16
JournalJournal of Mathematical Analysis and Applications
Volume141
Issue number1
DOIs
StatePublished - Jul 1989

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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