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Abstract  

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y i Y:= {y i } i∈ℤ such that for x ∈ [y i , y i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each nN(Y) we construct a trigonometric polynomial P n of order ≦ n, changing its sign at the same points y i Y as f, and

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {f - P_n } \right\| \leqq c(s)\omega _3 \left( {f,\frac{\pi } {n}} \right),$$ \end{document}
where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω 3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.

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Analysis Mathematica
Authors:
G. A. Dzyubenko
,
J. Gilewicz
, and
I. A. Shevchuk

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