Rate of convergence for generalized Szász operators

The present papers deals with the general integral modification of the Szász -Mirakyan operators having the weight functions of Baskakov basis functions. Here we estimate the rate of convergence for functions having derivatives of bounded variation.


Introduction
In the year 1983, Prasad et al [3] considered the integral modification of the well known Szász-Mirakyan operators by taking the weight functions of Baskakov operators and established some direct results in ordinary and simultaneous approximation. It was observed that there are some mistakes in [3], which were later improved by Gupta [1]. The general integral modification of the Szász-Mirakyan operators to approximate Lebesgue integrable functions on the interval [0, ∞), with weight functions of Baskakov basis functions can be defined as where n ∈ N , r ∈ N 0 , n > r and the Szász and Baskakov basis functions are defined as The family of operators G n,r ( f, x) is linear and positive. In the recent years the rate of convergence for functions of bounded variation is an active area of research, several researchers have studied in this direction, we refer some of the important papers in this area as [4][5][6] and [7] etc. Here we extent the studies and in the present paper, we estimate the rate of convergence for functions having derivatives of bounded variation. We have also studied the simultaneous approximation and in the end we mentioned an open problem for the readers.

Auxiliary results
Lemma 1 Let the mth order moment be defıned as then for n > r + m + 2, we have the following recurrence relation For all r, m ∈ N 0 and x ∈ [0, ∞), one can easily obtain from the recurrence formula that Proof Using the identities Now integrating by parts the last integral, we get The moments can be obtained easily by using the above recurrence relation, keeping in mind the fact that ∞ k=0 s n,k (x) = 1 and ∞ 0 b n,k (t)dt = 1 n−1 , we omit the details.
Remark 1 For n sufficiently large, C > 1 and x ∈ (0, ∞), it can be seen from Lemma 1, that Remark 2 By using Cauchy-Schwarz inequality, it follows from Remark 1, that for n sufficiently large , C > 1 and x ∈ (0, ∞) Lemma 2 Let x ∈ (0, ∞) and C > 1, then for n sufficiently large, we have Proof The proof of the above lemma follows easily by using Remark 1. For instance, for the first inequality for n sufficiently large and 0 ≤ y < x, we have The proof of the second inequality follows along the similar lines.

Lemma 3 Suppose f is s times differentiable on
for some positive integer q as t → ∞. Then for any r, s ∈ N 0 and n > max{q, r + s + 1}, we have Proof Using the identities One can observe that even in case k = 0, the above identities are true with the conditions that b n+1,negative (x) = 0 and s n,negative (x) = 0, thus applying (2.3), we Using (2.4), and integrating by parts we have which means that the identity is satisfied for s = 1. Let us suppose that the result holds for s i.e., Integrating by parts the last integral we get Thus we obtain that the result holds for s + 1, hence by the mathematical induction the proof of the lemma is complete.

Rate of convergence
By D B q (0, ∞), (where q is some positive integer) we mean the class of absolutely continuous functions f defined on (0, ∞) satisfying the following conditions: (i) f (t) = O(t q ), t → ∞ (ii) the function f has the first derivative on the interval (0, ∞) which coincide a.e. with a function which is of bounded variation on every finite subinterval of (0, ∞). It can be observed that for all functions f ∈ D B q (0, ∞) we can have the representation Theorem 1 Let f ∈ D B q (0, ∞),q > 0 and x ∈ (0, ∞). Then for C > 1 and n sufficiently large, we have where b a f (x) denotes the total variation of f x on [a, b], and f x is defined by Proof Using the mean value theorem, we can write Also, using the identity where Obviously, we have Thus, using above identities, we can write Also, it can be verified that and

(3.4)
Applying Remark 2 and Lemma 1, in ( 3.4), we have (3.5) In order to complete the proof of the theorem it suffices to estimate the terms A n,r ( f, x) and B n,r ( f, x). Applying integration by parts and Lemma 2 with Corollary 1 Let f (s) ∈ D B q (0, ∞), q > 0 and x ∈ (0, ∞). Then for C > 1 and for n sufficiently large, we have where b a f (x) denotes the total variation of f x on [a, b], and f x is defined by Remark 3 For every n ∈ N, q ∈ (0, 1) , the q analogue of the operators (1.1) for the case r = 0, can be defined as and b q n,k (t) := n + k − 1 k q q k(k−1)/2 t k (1 + t) n+k q for x ∈ [0, ∞) and for every real valued continuous function f on [0, ∞) . We use the following notations of q calculus (see [2] and references therein): For n ∈ N, the q-integers, q-factorial and q-binomial coefficients are defined as [n] q := 1 − q n 1 − q , [n] q ! := [n] q [n − 1] q · · · [1] q , n = 1, 2, . . . 1, n = 0 and n k q The q exponential function is defined as where (1 + x) ∞ q = ∞ j=0 (1 + q j x) and q-improper integral is defined as provided the sums converge absolutely.
In [2], the authors have studied some other approximation properties on these q operators, but we observe that the results analogous to the present paper are not possible for such q operators even for special value i.e., r = 0. This can be considered as an open problem.