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
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:84978829298
SN - 0021-9045
VL - 210
SP - 1
EP - 22
JO - Journal of Approximation Theory
JF - Journal of Approximation Theory
ER -