The order of comonotone approximation of differentiable periodic functions

摘要

Let Δ(1)(Y ) be a set of all 2 -periodic functions f that are continuous on the real axis R and change their monotonicity at various fixed points yi ∈ - ; ); i = 1; :::; 2s; s ∈ N (i.e., there is a set Y := {yi}i∈? of points yi = yi+2s + 2 on R such that f are nondecreasing on yi; yi?1 if i is even, and nonincreasing if i is odd). In the article, a function fY = f 2 C(1) ∩ Δ(1)(Y ) has been constructed suchthatlimn→∞supnEn1fω4f′π/n=∞,documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ underset{nto infty }{lim}sup frac{n{E}_n^{(1)}(f)}{omega_4left({f}^{prime },pi /nright)}=infty, $$end{document}where En1fdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {E}_n^{(1)}(f) $$end{document} is the error of the best uniform approximation of the function f ∈ Δ(1)(Y) by trigonometric polynomials of order n ∈ N, which also belong to the set Δ(1)(Y ), and 4(f’; ?) is the 4-th modulus of smoothness of the function f’. So, for a certain constant c, the inequality En1f≤cnω3f′π/ndocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {E}_n^{(1)}(f)le frac{c}{n}{omega}_3left({f}^{prime },pi /nright) $$end{document} is the best with respect to the order of the modulus of smoothness.