Yet another look at positive linear operators, q-monotonicity and applications

JOURNAL OF APPROXIMATION THEORY

卷:210

Article

Kopotun, K. A.¹  Leviatan, D.²  Prymak, A.¹  Shevchuk, I. A.³ 

[1] Univ Manitoba, Dept Math, Winnipeg, MB R3T 2N2, Canada

[2] Tel Aviv Univ, Raymond & Beverly Sackler Sch Math Sci, IL-69978 Tel Aviv, Israel

[3] Taras Shevchenko Natl Univ Kyiv, Fac Mech & Math, UA-01033 Kiev, Ukraine

关键词: Positive linear operators;    Degree of approximation;    Jackson-type estimates;    Moduli of smoothness;    Gavrea's operator;    Bernstein-Durrmeyer-Lupas polynomials with ultraspherical weights;   

DOI  :  10.1016/j.jat.2016.06.001

来源: Elsevier

【 摘 要 】

For each q is an element of N-0, we construct positive linear polynomial approximation operators M-n that simultaneously preserve k-monotonicity for all 0 <= k <= q and yield the estimate vertical bar f(x) - M-n(f, x)vertical bar <= c omega(phi lambda)(2) (f,n(-1)phi(1-lambda/2)(x) (phi(x) + 1/n)(-lambda/2)), for x is an element of [0, 1] and lambda is an element of [0, 2), where phi(x) := root x(1-x) and omega(psi)(2) is the second Ditzian-Totik modulus of smoothness corresponding to the step-weight function psi. In particular, this implies that the rate of best uniform q-monotone polynomial approximation can be estimated in terms of omega(psi)(2) (f, 1/n). (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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