Ukrainian Mathematical Journal

, Volume 48, Issue 3, pp 367–376 | Cite as

Copositive pointwise approximation

  • G. A. Dzyubenko
Article

Abstract

We prove that if a functionfC (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
$$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$
, 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.

Keywords

Polynomial Kernel Pointwise Approximation Algebraic Polynomial Spline Approximation Nonincreasing Function 

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© Plenum Publishing Corporation 1997

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  • G. A. Dzyubenko

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