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Almost Coconvex Approximation of Continuous Periodic Functions

  • G. A. DzyubenkoEmail author
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If a 2 𝜋 -periodic function f continuous on the real axis changes its convexity at 2s, s ∈ ℕ, inflection points yi : 𝜋 ≤ y2s< y2s−1< ... < y1< 𝜋 and, for all other i ∈ ℤ, yi are periodically defined, then, for any natural n ≥ Nyi, we can find a trigonometric polynomial Pn of order cn such that Pn has the same convexity as f everywhere except, possibly, small neighborhoods of the points yi : (yi 𝜋/n, yi + 𝜋/n) and, moreover,
$$ \left\Vert f-{P}_n\right\Vert \le c(s){\omega}_4\left(f,\uppi /n\right), $$

where Nyi is a constant that depends only on mini=1,...,2s{yi− yi+1}, c and c(s) are constants thatdepend only on s, 𝜔4(f, ·) is the fourth modulus of smoothness of the function f, and ‖⋅‖is the max-norm.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics, Ukrainian National Academy of SciencesKyivUkraine

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