Comonotone approximation of periodic functions |
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Authors: | G A Dzyubenko M G Pleshakov |
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Institution: | 1. International Mathematical Center, National Academy of Sciences of Ukraine, Kharkov, 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, on y i , y i?1], f is nondecreasing if i is odd and nonincreasing 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 $$ \left\| {f - P_n } \right\| \leqslant c\left( s \right)\omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω 2(f, ·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm. |
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