Estimates of positive linear operators in terms of second-order moduli

Abstract

We estimate the constants related with the direct result for positive linear operators which preserves linear functions. The estimates are presented for the modulus of smoothness ω ϕ 2 ( f , h), where the weight ϕ is a function such that ϕ 2 is concave.

Asymptotic and Non-asymptotic Results in the Approximation by Bernstein Polynomials

“…Whereas no specific values for K 1 have been provided yet, different authors completed statement (4) by showing specific values for the constant K 2 . In this regard, Adell and Sangüesa [1] gave K 2 = 4, Gavrea et al [5] and Bustamante [2] provided K 2 = 3, and finally, Pȃltȃnea [7] proved the validity of K 2 = 2.5, this being the best result up to date and up to our knowledge. This notwithstanding, if additional smoothness conditions on f are added, then the second inequality in (4) may be valid for values of K 2 smaller than 2.5.…”

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

“…We say that a function f ∈ C[0, 1] is q-monotone if ∆ q δ (f, x) ≥ 0 for all δ > 0, and denote the set of all q-monotone (continuous) functions by ∆ (q) . In particular, ∆ (0) , ∆ (1) and ∆ (2) are, respectively, the classes of all nonnegative, nondecreasing and convex functions from C[0, 1]. We also remark that, for q ≥ 3, f ∈ C[0, 1] is q-monotone if and only if f ∈ C q−2 (0, 1) and f (q−2) is convex in (0, 1).…”

Direct and strong converse theorems for a general sequence of positive linear operators

“…In [6] Felten verified that the Ditzian estimate holds for a large class of weight functions ϕ. Felten's results were improved in [2] as follows. Let L : C[0, 1] → C[0, 1] be a positive linear operator which preserves linear functions.…”

“…Let q α ∈ (0, 1) and k α be as in the first condition in (2). There exists an integer p such that 8C 4 q p α 1.…”