Comonotone approximation of periodic functions |
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Authors: | G. A. Dzyubenko M. G. Pleshakov |
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Affiliation: | 1. International Mathematical Center, National Academy of Sciences of Ukraine, Kiev, Ukraine 2. Saratov State University, Saratov, Russia
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Abstract: | Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y i } i∈? of points y i = y i+2s + 2π on ? such that f is nondecreasing on [y i ,y i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P n of order ≤ n changing its monotonicity at the same points y i ∈ Y as f and such that $$ parallel f - P_n parallel leqslant c(s) omega _2 left( {f,frac{pi } {n}} right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω2(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm. |
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