Comonotone Jackson's Inequality |
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Authors: | M G Pleshakov |
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Institution: | Saratov State University, Rabotchaja 34-1, Saratov, 410026, Russia |
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Abstract: | Let 2s points yi=−πy2s<…<y1<π be given. Using these points, we define the points yi for all integer indices i by the equality yi=yi+2s+2π. We shall write fΔ(1)(Y) if f is a 2π-periodic continuous function and f does not decrease on yi, yi−1], if i is odd; and f does not increase on yi, yi−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved.
1. If fΔ(1)(Y), then for each nonnegative integer n there is a trigonometric polynomial τn(x) of order n such that τnΔ(1)(Y), and |f(x)−πn(x)|c(s) ω(f; 1/(n+1)), x
, where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s. |
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Keywords: | comonotone approximation polynomial approximation |
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