-Approximation by -Kantorovich Operators

For a new -Kantorovich operator we establish direct approximation theorems in the space via Ditzian-Totik modulus of smoothness of second order. 1. Introduction Let . Then for each non-negative integer , the -integer and the -factorial are defined by , , and , . For integers , the -binomial coefficient is defined by Further, we set for and . Following Phillips [1], the operators defined by are called -Bernstein operators. They quickly gained the popularity and were studied widely by a number of authors. A survey of the obtained results and references on the subject can be found in [2]. The -Bernstein operators are not defined for . For this reason we introduce the following operators based on -integers: For , we recover the well-known Kantorovich operators [3]. In [4, 5] other Kantorovich type operators involving -integers are considered, which are defined with the aid of -Riemann integral (see [6]), and their rate of convergence are studied for only continuous functions on . The aim of the paper is to establish direct approximation theorems for (4) in the space , . To describe our results, we will give the definitions of Ditzian-Totik modulus of smoothness of second order and the corresponding K-functionals (cf. [7]). For , , and , , we set where and means that is differentiable such that is absolutely continuous in every interval . It is known from [7] that , and are equivalent; that is, there exists such that Here denotes a positive constant independent of and , but it is not necessarily the same in different cases. Finally, we define the modulus of continuity of by The paper is organized as follows. In Section 2 some auxiliary results are established and in Section 3 we give the main results, establishing direct approximation theorems for (4) in , . 2. Auxiliary Results In the sequel we need some lemmas. Lemma 1. Let , , be defined by (4) and . Then , and . Proof. By (4), we obtain Recall some properties of the -Bernstein operators (see [1]): Then, by (9) and (10), we find and But implies that for . Hence, by (10), Further, by (9), (10) and (12), we have because Lemma 2. Let and let be given by (4). Then (i) , where , and there exists such that for all ;(ii) , where . Proof. By Riesz-Thorin theorem (see, e.g., [8]), we separate the proof for and . Case？？1 ( ). Because the -binomial coefficients are increasing functions of , we get for the polynomials and of degree that Now let and . Then, in view of , , we obtain Using [7, Theorem 8.4.8, page 108] translated from to , we find that On the other hand , , implies that , . Hence, by (4) and (18), we get