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Stechkin-Type Estimate for Nearly Copositive Approximations of Periodic Functions

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Ukrainian Mathematical Journal Aims and scope

Assume that a continuous 2𝜋 -periodic function f defined on the real axis changes its sign at 2s, s ∈ ℕ, points yi : −𝜋 ≤ y2s < y2s−1 < ... < y1 < 𝜋 and that the other points yi, i ∈ ℤ, are defined by periodicity. Then, for any natural n > N(k, yi), where N(k, yi) is a constant that depends only on k ∈ ℕ and mini=1,...,2s{yi − yi+1}, we construct a trigonometric polynomial Pn of order ≤ n, which has the same sign as f everywhere, except (possibly) small neighborhoods of the points yi : (yi − π/n, yi + π/n), Pn(yi) = 0, i ∈ ℤ, and in addition, ‖f − Pn‖ ≤ c(k, s)ωk(f, π/n), where c(k, s) is a constant that depends only on k and s, 𝜔k(f, ·) is the k th modulus of smoothness of f, and ∥·∥ is the max-norm.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    G. A. Dzyubenko

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  1. G. A. Dzyubenko
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Correspondence to G. A. Dzyubenko.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 5, pp. 628–634, May, 2020.

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Dzyubenko, G.A. Stechkin-Type Estimate for Nearly Copositive Approximations of Periodic Functions. Ukr Math J 72, 722–729 (2020). https://doi.org/10.1007/s11253-020-01812-y

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