tailieunhanh - On the order of weighted approximation by positive linear operators

An estimation of approximation of continuous functions by positive linear operators in weighted norm using the weighted modulus of continuity is established. Application of the main result to the known Gadjiyev-Ibragimov operators is given. | Turk J Math 36 (2012) , 113 – 120. ¨ ITAK ˙ c TUB doi: On the order of weighted approximation by positive linear operators T¨ ulin Co¸skun Abstract An estimation of approximation of continuous functions by positive linear operators in weighted norm using the weighted modulus of continuity is established. Application of the main result to the known Gadjiyev∗ -Ibragimov operators is given. Key Words: Positive linear operators, weighted spaces, weighted modulus of continuity, Korovkin type theorems, Gadjiev-Ibragimov operators 1. Introduction and preliminaries Let m be a natural number and B2m = B2m (0, ∞) be the space of all functions f defined on the semi-axis [0, ∞) satisfying the inequality |f(x)| ≤ Mf (1 + x2m ), where Mf is a positive constant depending on functions f only. Obviously, B2m is the linear normed space with the norm |f(x)| . 2m 1 x≥0 + x f 2m = sup The subspace of all continuous functions belonging to B2m will be denoted by C2m := C2m (0, ∞), and also define 0 C2m := {f ∈ C2m : lim x→∞ f(x) = Kf 0 , we can find x0 > 0 such that | f(x + h) − Kf | x0 . Also, there exist δ > 0 such that the inequality sup 0≤x≤x0, |h|≤δ |f(x + h) − f(x)| 0 . for any function f ∈ C2m Since ω(f, δ)2m tends to zero as δ → 0 , it may be used for the estimate of degree of approximation of 0 functions in C2m by . We will use the sequences of . introduced in [14] called the Gadjiev-Ibragimov operators (see, for example, [16]). These operators were studied in different papers, for example, in references [2], [6], [7], [13], [15] and [16]. In particular, in [14] authors established the estimate for |Ln (f,x)−f(x)| , (1+x2 )α where α ≥ 3, (Ln ) are Gadjiev- Ibragimov operators, f ∈ C20 and x ∈ [0, γn ], lim γn = ∞ . In [14] the estimate for the same ratio is stated as n→∞ an “open problem”, if α ∈ [1, 3). Our main theorem, which is proved in the space C2m in the last section of this paper, strengthens the above result for m = 1 . This