GLOBAL APPROXIMATION BY MODIFIED BASKAKOV TYPE OPERATORS

In the present paper, we prove a global direct theorem for the modified Baskakov type operators in terms of so called DitzianTotik modulus of smoothness.


Introduction
Motivated by the integral modification of Bernstein polynomials by Durrmeyer [3], Sahai and Prasad [6] first defined and studied modified Baskakov operators.Sinha et al. [7] improved and corrected the results of [6].Recently the author [4], introduced another modification of Baskakov operators by taking the weight function of Beta operators on where B(k + 1, n) being the Beta function given by k!(n − 1)!/(n + k)!.
In [4], the author has obtained only local direct theorems in simultaneous approximation, as the operators defined by (1.1) give better approximation than the earlier integral modification of Baskakov operators Research supported by Council of Scientific and Industrial Research, India under award no.9/143(163)/91-EMR-1.

V. Gupta
studied in [5], [6] and [7] etc., this motivated us to extend the results of [4] to the whole interval [0, ∞) and we study a global result for the operators (1.1).
By L r 1 [0, ∞), we denote the class of functions g given by and m are constants depending on g}.
We may remark that L r p [0, ∞) is not contained in L r 1 [0, ∞).Following [2], the modulus of smoothness of f is given by where This modulus of smoothness is equivalent to the modified k-functional (see e.g.[2]) given by In [4] the author was not able to obtain global results.In the present paper, we prove a global direct theorem in simultaneous approximation for the operators (B n f )(x) defined by (1.1) in terms of Ditzian-Totik modulus of second order.
Throughout the paper we denote by C the positive constants not necessarily the same at each occurrence.

Auxiliary results
In this section, we shall give certain definitions and lemmas which will be used in the sequel.
For every n ∈ N and n > (r + 1) we have and there holds the recurrence relation: The proof of this lemma easily follows along the lines of [6], [7] using From the above lemma, we have where q i,m,n (x) and s i,m,n (x) are polynomials in x of fixed degree with coefficients that are bounded uniformly for all n. .
Proof: By using Leibnitz theorem, we have Again, by the use of Leibnitz theorem, we have b Hence, On integrating r times by parts, we get the required result.
We see that the operators defined in (2.3) by B (r) To make the operators positive we introduce the operator where D and I are differentiation and integration operators respectively.Therefore we define the operator by The operators B n,r are positive and the estimation (B n f ).Using (2.1), we can easily prove that for n > (r + 1), Making use of Riesz-Thorin theorem, we get Corollary 2.3.For every m ∈ N 0 , n > (r + 2m + 1) and x ∈ [0, ∞) we have where the constant C is independent of n.For fixed x ∈ [0, ∞) we obtain the estimate (2.5) follows from (2.2) along the lines of [5], (2.6) immediately follows from (2.5).

Lemma 2.4. Let
where the constant C is independent of n. and

V. Gupta
Making use of these two identities and (2.1) we get For the two monomials e 0 , e 1 and x ∈ [0, ∞), n → ∞ we obtain by direct computation (2.8) Lemma 2.5.For H n (u) given by , where C is independent of n and u.
The proof of the above lemma easily follows by using (2.1) along the lines of [1, Lemma 5.2].

Direct result
where the constant C is independent of n.