Pointwise estimation of an almost copositive approximation of continuous functions by algebraic polynomials
Abstract
In the case where a function continuous on a segment f changes its sign at s points yi:1<ys<ys−1<...<y1<1, for any n∈N greater then a constant N(k,yi) that depends only on k∈N and \min, we determine an algebraic polynomial P_n of degree \leq n such that: P_n has the same sign as f everywhere except possibly small neighborhoods of the points y_i: ((y_i \rho_n(y_i), y_i + \rho_n(y_i)),\quad \rho_n(x) := 1/n2 + \sqrt{1 - x^2}/n, P_n(y_i) = 0 and | f(x) P_n(x)| \leq c(k, s)\omega_k(f, \rho_n(x)),\quad x \in [ 1, 1], where c(k, s) is a constant that depends only on k and s and \omega k(f, \cdot ) is the modulus of continuity of the function f of order k.Downloads
Published
25.05.2017
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Section
Research articles
How to Cite
Dzyubenko, H. A. “Pointwise Estimation of an Almost Copositive Approximation of Continuous Functions by Algebraic Polynomials”. Ukrains’kyi Matematychnyi Zhurnal, vol. 69, no. 5, May 2017, pp. 641-9, https://umj.imath.kiev.ua/index.php/umj/article/view/1723.