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Interpolated Estimate for Copositive Approximations by Algebraic Polynomials

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Ukrainian Mathematical Journal Aims and scope

Under the condition that a function f continuous on [−1, 1], changes its sign at s points yi, − 1 < ys < ys − 1 < …y1 < 1, for each n ∈ ℕ greater than some constant N that depends only on

$$ \underset{i=0,\dots, s}{\min}\left\{{y}_i-{y}_i+1\right\},\kern0.5em {y}_{s+1}:= -1,\kern0.5em {y}_0:= 1, $$

we construct an algebraic polynomial Pn of degree ≤ n such that Pn has the same sign as f on [−1, 1]. In particular, Pn(yi) = 0, i = 1, …, s, and \( \left|f(x)-{P}_n(x)\right|\le c(s){\omega}_2\left(f,\sqrt{1-{x}^2}/n\right),x\in \left[-1,1\right], \) where c(s) is a constant that depends only on s and ω2(f, ⋅) is the modulus of smoothness of the second order for the function f. Note that this estimate was established by DeVore for the unconstrained approximation. It is interpolating at ±1 and it is impossible to replace ω2 with ωk, k > 2, even for the unconstrained approximation.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, 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. 74, No. 4, pp. 496–506, April, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i4.7103.

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Dzyubenko, G.A. Interpolated Estimate for Copositive Approximations by Algebraic Polynomials. Ukr Math J 74, 563–574 (2022). https://doi.org/10.1007/s11253-022-02083-5

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