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Stechkin-Type Estimate for Nearly Copositive Approximations of Periodic Functions
Ukrainian Mathematical Journal ( IF 0.5 ) Pub Date : 2020-10-01 , DOI: 10.1007/s11253-020-01812-y
G. A. Dzyubenko

Assume that a continuous 2𝜋 -periodic function f defined on the real axis changes its sign at 2s, s ∈ ℕ, points yi : −𝜋 ≤ y2s 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.

中文翻译:

周期函数的近似正近似的 Stechkin 型估计

假设定义在实轴上的连续 2𝜋 -周期函数 f 在 2s 处改变其符号,s ∈ ℕ,点 yi : −𝜋 ≤ y2s N(k, yi),其中 N(k, yi) 是一个常数,取决于仅在 k ∈ ℕ 和 mini=1,...,2s{yi − yi+1} 上,我们构造了一个 ≤ n 阶的三角多项式 Pn,它在任何地方都与 f 具有相同的符号,除了(可能)小邻域点 yi : (yi − π/n, yi + π/n), Pn(yi) = 0, i ∈ ℤ, 此外, ‖f − Pn‖ ≤ c(k, s)ωk(f, π /n),其中 c(k, s) 是仅取决于 k 和 s 的常数,𝜔k(f, ·) 是 f 的第 k 个平滑模数,∥·∥ 是最大范数。
更新日期:2020-10-01
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