QUANTITATIVE ESTIMATES FOR JAIN-KANTOROVICH

By using given arbitrary sequences, βn > 0, n ∈ N with the property that limn→∞nβn = 0 and limn→∞βn = 0, we give a Kantorovich type generalization of Jain operator based on the a Poisson disrtibition. Fristly we give the quantitative Voronovskaya type theorem. Then we also obtain the Grüss Voronovskaya type theorem in quantitative form .We show that they have an arbitrary good order of weighted approximation.


EM RE DEN • IZ
Recently, Agratini [3] studied class of integral type positive linear operators of P [ ] n and obtained some approximation properties of them in weighted spaces.Also, some authors studied generalizations of Jain's operators ( [10], [13], [16] and [17]).Now, we de…ne and investigate Kantorovich variant of P [ ] n operator , in order to obtain an approximation process for spaces of locally integrable functions on unbounded interval, replacing the sample values f (k=n) by the mean values of f in the intervals k n ; k+1 n as follows: De…nition 1.For 2 [0; 1) where The Kantorovich method was applied to many generalizations of the Bernstein polynomials like for example Szász-Mirakyan, Baskakov and other operators.A recent contribution in this direction was given in [4].We note that, P. L. Butzer [5] introduced and studied Szasz-Mirakyan-Kantorovich operators de…ned by for f 2 L 1 (0; 1), the space of integrable functions on unbounded interval [0; 1): In this paper we study some approximation properties of the sequence of linear positive operators given by (1.4) in a weighted space.
The structure of the paper is as follows.In the second section, we calculate some moment of our operator in De…nition 1.In the third section, a Voronovskaya type theorem in quantitative form is obtained as well.In the fourth section, we also give a Grüss Voronovskaya type theorem in quantitative form.In the last section, some weighted approximation theorems are presented.

Moments of the Operators S [ ] n
We begin with the following lemma which is necessary to prove the main result.Taking in view Lemma 1 in [2] has been established the following moments: Lemma 1.Let e j ; j 2 N [ f0g ; be the j-th monomial, e j (t) = t j : For the operators de…ned by (1.2) (see also [14,Eq.(2.11)]) the moments are as follows: n (e 0 ) (x) = 1: With a simple calculation, we obtain that Similarly, for j 4; the proof can be done.

Voronovskaya Theorems
Let B x 2 [0; 1) be the set of all functions f de…ned on [0; 1) satisfying the condition jf (x)j M f 1 + x 2 with some constant M f , depending only on f; but independent of x.B x 2 [0; 1) is called weighted space and it is a Banach space endowed with the norm 1+x 2 is …nite.We know that usual …rst modulus of continuity ! ( ) does not tend to zero, as !0; on in…nite interval.Thus we use weighted modulus of continuity (f; ) de…ned on in…nite interval [0; 1) (see [12]).Let Now some elementary properties of (f; ) are collected in the following Lemma.
iii) For each > 0; From the inequality (3.1) and de…nition of (f; ) we get for every f 2 C x 2 [0; 1) and x; t 2 [0; 1) : Next, we give the quantitative Voronovskaya type theorem in weighted spaces, which states the following: x 2 [0; 1) and 0 n < 1: Then, we have where n !0; depending on n ; as n ! 1 and C is a positive constant.
Proof.By the local Taylor's formula there exist lying between x and y such that where and h is a continuous function which vanishes at 0: Applying the operator S [ n ] n to above equality, we obtain the equality also we can write that To estimate last inequality using the inequality (3.2) and the inequality j xj jy xj ; we can write that jh ( ; x)j 1 + (y x) We deduce that ) Using Lemma 1 and calculating with simple, we have where C is a positive constant.Thus we have desired result.

Remark 1. It is seen that S [ ]
n does not form an approximation process.In order to transform it into an approximation process, the constant will be replaced by a number n 2 [0; 1) and also lim n!1 n n = 0: The following estimate is Voronovskaya type asymptotic formula.

Grüss Type Approximation
Let us to prove the following result called by us Grüss-Voronovskaya type theorem in quantitative form (see [9]).
Theorem 2. Suppose that the …rst and second derivative f 0 ; g 0 ; f 00 ; g 00 and (f g) 00 exist at a point x 2 [0; 1) ; we have where n and n !0; depending on n ; n !1: Proof.For x 2 [0; 1) ; we have Thus we can write nS n S [ n ] n (g; x) g (x) From Theorem 1, we have On the other hand, we can write n f 00 ( ) (t x) 2 ; x : Therefore, we have where is a number between t and x.Case 1: t < < x; : Thus, we obtain for two cases of that  Thus the proof is completed.

Weight Approximation
Now, in this section we give some weight approximation theorems for the functions which belong to weighted space C k x 2 [0; 1) by S [ ] n operators.For details of proofs see [2] and [8].Now, we give the following theorem to approximation all functions in C x 2 [0; 1) : This type of results is given in Gadjiev et al. [7] for locally integrable functions. )

Theorem 3 . 0 STheorem 4 . 1 S
If f 2 C k x 2 [0; 1).then the inequality sup x a su¢ ciently large n; where K is a constant.For each f 2 C k x 2 [0; 1), we have lim n!