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* The second author is supported by the Alexander-von-Humboldt Foundation. He is on leave from: Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China.
The best constant problem for Bernstein operators with respect to
the second modulus of smoothness is considered. We show that for any
1/2 ≤ a < 1, there is an N(a) ∈ N such that for n ≥ N(a),
1−a≤k, n≤a, sup | Bn (f, k/n) − f(k/n) | ≤ cω2(f, 1/√n),
where c is a constant,0 < c < 1.
In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and...
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