New Moduli of Smoothness: Weighted DT Moduli Revisited and Applied

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We introduce new moduli of smoothness for functions (Formula Presented), that have an (r-1)st locally absolutely continuous derivative in (-1,1), and such that φrf(r) is in Lp[-1,1], where (Formula Presented). These moduli are equivalent to certain weighted Ditzian–Totik (DT) moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in $$L_p[-1,1]$$Lp[-1,1] (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems, thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.

Original languageEnglish
Pages (from-to)129-159
Number of pages31
JournalConstructive Approximation
Issue number1
StatePublished - 1 Aug 2015


  • Approximation by polynomials in the L-norm
  • Degree of approximation
  • Jackson-type estimates
  • Moduli of smoothness


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