Abstract
We prove that if a functionf ∈C (1) (I),I: = [−1, 1], changes its signs times (s ∈ ℕ) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ℕ, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality
, where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k − 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,x ∈I, and |f(x) −P n (x)| ≤c(k)ω k (f;n −2 +n −1 √1 −x 2),x ∈I.
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Dzyubenko, G.A. Copositive pointwise approximation. Ukr Math J 48, 367–376 (1996). https://doi.org/10.1007/BF02378527
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DOI: https://doi.org/10.1007/BF02378527