Copositive approximation of periodic functions |
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Authors: | G A Dzyubenko J Gilewicz |
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Institution: | (1) International Mathematical Center of NAS of Ukraine, 01601, Tereschenkivs’ka str., 3, Kyiv, Ukraine;(2) Toulon University and Centre de Physique Théorique, CNRS — Luminy, Case 907, 13288 Marseille Cedex 09, France |
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Abstract: | Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y
i
∈ Y:= {y
i
}
i∈ℤ such that for x ∈ y
i
, y
i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P
n
of order ≦ n, changing its sign at the same points y
i
∈ Y as f, and where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω
3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.
This work was done while the first author was visiting CPT-CNRS, Luminy, France, in June 2006. |
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Keywords: | and phrases" target="_blank"> and phrases copositive polynomial approximation uniform estimates |
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