We say that a function f ∈ C[a, b] is q-monotone, q ≥ 2, if f ∈ Cq-2(a, b), i.e., belongs to the space of functions with (q -2)th continuous derivative in (a, b), and f(q-2) is convex in this space. Let f be continuous and 2𝜋-periodic. Assume that it changes its q-monotonicity finitely many times in [-𝜋, 𝜋]. We are interested in estimating the degree of approximation of f by trigonometric polynomials, which are co-q-monotone with this function, namely, trigonometric polynomials that change their q-monotonicity exactly at the points where f does. These Jackson-type estimates are valid for piecewise monotone (q = 1) and piecewise convex (q = 2) approximations. However, we prove, that no estimates of this kind are valid, in general, for the co-q-monotone approximation with q ≥ 3.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 5, pp. 662–675, May, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i5.7081.
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Leviatan, D., Motorna, O.V. & Shevchuk, I.A. No Jackson-Type Estimates for Piecewise q-Monotone, q≥3, Trigonometric Approximation. Ukr Math J 74, 757–772 (2022). https://doi.org/10.1007/s11253-022-02099-x
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DOI: https://doi.org/10.1007/s11253-022-02099-x