Abstract
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
Case 2 (). Due to (9) and (10), we have
Lemma 3. If is defined by (4), , and , then where satisfies that there exists such that for all .
Proof. Consider the following cases.
Case 1 (). Because of [7, Lemma 9.6.1, page 140], we have . Then, by (4),
But
imply
Hence, by (22),
where is either or such that
Using the method in [7, pages 146-147], let
Then
For we prove that
To obtain (29), we need some useful estimations:
Indeed, for we have . Hence, by (15),
Further, for we have
taking into account (15) and , because of .
Now let and . By (15), we have . Then, in view of (30), (17), [7, Lemma 9.4.4, page 128], (15) and (31), we obtain
For , by (10), (15) and (31), we have
We now define
Then , where , because (23), (30) and the definition of imply that
Hence, by (29), (28) and (15), we get
Applying the estimate (see [7, (9.6.14), page 147])
for and recalling that , we have
Substituting this estimate in (38) for and following the procedure given in [7, page 147], we obtain the statement of our lemma, because of (25).
Case 2 (). Let . The function is concave on ; therefore
(see [9, (4)]). Then, by Lemma 1,
where is the Hardy-Littlewood maximal function. It is known that ,. Hence, by (42),
But , because of (31). Thus
which completes the proof.
Lemma 4. If is a polynomial of degree at most , where , then is also a polynomial of degree at most .
Proof. It is sufficient to prove that is a polynomial of degree at most . By (4), we have Since is a polynomial of degree (see [1]), we obtain the assertion of the lemma.
3. Main Results
The direct approximation results are included in the following theorem.
Theorem 5. Let be arbitrary and let be given by (4). (i)If , , and satisfies that for all , then (ii)If satisfies that as , then
Proof. Let . By Taylor’s formula,
and Lemma 1, we obtain
Hence, for , we have
Using the estimate , (see [7, (a), page 135]), and Lemma 3, we get, in view of (50), that
On the other hand, the condition , , implies that and . Then . In conclusion
Finally, we follow the procedure given in [7, pages 118-119]. The definition of implies the existence of such that
Then, by (51), (53), (54) and (6), we get
Further, due to (54), (15) and (52), we have . Then, by [7, Theorem 7.3.1, page 84], we can choose to be the best th degree polynomial approximation in . Therefore, by (52) and (55), we obtain
Taking into account Lemma 4 and Theorem 8.4.8 of [7, page 108] translated from to and to , we obtain, using (56), that
Hence, due to Lemma 2, (53) and (6), we find
We introduce the operator
where . By Lemmas 1 and 2, we have and
Further, by (4) and (10), we obtain
Thus . Hence, in view of Taylor’s formula,
and (59), we get
But
Hence, by (63), (41), Lemma 1 and , we find
Now, from (59), (60), (65) and (64), we have
Taking the infimum on the right-hand side over all , and using the equivalence between and (see (7)), we obtain the statement of the theorem.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.