TY - JOUR

T1 - Yet another look at positive linear operators, q-monotonicity and applications

AU - Kopotun, K. A.

AU - Leviatan, D.

AU - Prymak, A.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - For each q∈N0, we construct positive linear polynomial approximation operators Mn that simultaneously preserve k-monotonicity for all 0≤k≤q and yield the estimate|f(x)−Mn(f,x)|≤cω2φλ(f,n−1φ1−λ/2(x)(φ(x)+1/n)−λ/2), for x∈[0,1] and λ∈[0,2), where φ(x):=x(1−x) and ω2ψ is the second Ditzian–Totik modulus of smoothness corresponding to the “step-weight function” ψ. In particular, this implies that the rate of best uniform q-monotone polynomial approximation can be estimated in terms of ω2φ(f,1/n).

AB - For each q∈N0, we construct positive linear polynomial approximation operators Mn that simultaneously preserve k-monotonicity for all 0≤k≤q and yield the estimate|f(x)−Mn(f,x)|≤cω2φλ(f,n−1φ1−λ/2(x)(φ(x)+1/n)−λ/2), for x∈[0,1] and λ∈[0,2), where φ(x):=x(1−x) and ω2ψ is the second Ditzian–Totik modulus of smoothness corresponding to the “step-weight function” ψ. In particular, this implies that the rate of best uniform q-monotone polynomial approximation can be estimated in terms of ω2φ(f,1/n).

KW - Bernstein–Durrmeyer–Lupaş polynomials with ultraspherical weights

KW - Degree of approximation

KW - Gavrea's operator

KW - Jackson-type estimates

KW - Moduli of smoothness

KW - Positive linear operators

UR - http://www.scopus.com/inward/record.url?scp=84978829298&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2016.06.001

DO - 10.1016/j.jat.2016.06.001

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AN - SCOPUS:84978829298

SN - 0021-9045

VL - 210

SP - 1

EP - 22

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

ER -