Pointwise Estimate for Linear Combinations of Bernstein–Kantorovich Operators

Abstract

For linear combinations of Bernstein-Kantorovich operators K n r f x , we give an equivalent theorem with ω

L p approximation by Bernstein-Kantorovich quasi-interpolants

“…Proof. Using Lemma 3.1 and Lemma 3.2, the proof of Theorem 3.1 is similar to that of Theorem 1 of [3]. The details are omitted.…”

“…In order to obtain faster convergence, various modifications of Bernstein-Kantorovich operator, linear combinations of Bernstein-type sequences [2,3] or a family of intermediate Bernsteintype operators were considered to accelerate this convergence [4,5] . In [5], Sablonnière introduced such last-mentioned intermediate operator K (r) n , (n ≥ r) between Bernstein-Kantorovich operator K n and Lagrange interpolation operator and called it Bernstein-Kantorovich quasiinterpolants.…”

Simultaneous approximation by combinations of Bernstein–Kantorovich operators

“…The main result in [7] is Theorem A. If r ∈ N, 0 λ 1 and 0 < α < 2r 2−λ , then for all f ∈ C[0, 1], we have…”

“…From Lemma 3.2 in [7], we can easily obtain Lemma 3.2. If r ∈ N, 0 < β < 2r, 0 t 1 8r and rt < x < 1 − rt, then…”

Approximation by Bézier type of Meyer–König and Zeller operators

“…Direct and inverse theorems have been investigated in some approximation processes (cf. [1][2][3][4][5][6][7][8][9][10][11][12]). …”