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Copositive pointwise approximation
Authors:G A Dzyubenko
Abstract:We prove that if a functionfC (1) (I),I: = ?1, 1], changes its signs times (s ∈ ?) 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 ∈ ?, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iffC (I) andf(x) ≥ 0,xI then, for anynk ? 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,xI, and |f(x) ?P n (x)| ≤c(k k (f;n ?2 +n ?1 √1 ?x 2),xI.
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