Comonotone approximation by hybrid polynomials

D. Leviatan*, M. V. Shcheglov, I. A. Shevchuk

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let f(x):=g(x)+ax, where g∈C˜, the space of continuous 2π-periodic functions and a∈R. Denote ‖g‖:=maxx∈R⁡|g(x)|, and let ω(f,t), denote the modulus of continuity of f. Let Tn be the set of trigonometric polynomials Tn of degree <n. We call Qn(x):=Tn(x)+ax, a∈R, a hybrid polynomial. If f is monotone, then g was called, by Salem and Zygmund, of monotone type. If f has an even number of extremal points in (−π,π], then we estimate inf⁡{‖f−Qn‖:Qns.t.f(x)Qn(x)≥0, a.e. in R}, the error of its best comonotone approximation, in the uniform norm, by hybrid polynomials. We obtain Jackson-type estimates for the approximation of f by hybrid polynomials for a wide class of functions f. We also show cases where such estimates are invalid.

Original languageEnglish
Article number127286
JournalJournal of Mathematical Analysis and Applications
Volume529
Issue number2
DOIs
StatePublished - 15 Jan 2024

Keywords

  • Comonotone approximation by hybrid polynomials
  • Degree of approximation
  • Jackson-type estimates

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