Some Classes of Function Spaces, Their Properties, and Their Applications 2014View this Special Issue
Research Article | Open Access
Simultaneous Approximation for Generalized Srivastava-Gupta Operators
We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval and estimate the rate of convergence for functions having derivatives of bounded variation. Also we present simultenaous approximation by new operators in the end of the paper.
To approximate integrable functions on the interval , Srivastava and Gupta  introduced a general sequence of linear positive operators as follows:for a function , where is the class of locally integrable functions defined on and satisfying the growth condition
The general sequence of operators has many interesting properties in approximation theory, which is an interesting area of research in the present era, and several researchers have studied these operators; we can mention some important studies on these operators (see [1–3]). In , author introduced King and Stancu type generalization of Srivastava-Gupta operators and presented some direct results. Also, Verma and Agrawal  introduced a new generalization of Srivastava-Gupta operators and studied the rate of convergence for the functions having the derivatives of bounded variation (BV). The rate of convergence for the functions having the derivatives of (BV) is an active area of research and many researchers studied this direction. We refer the readers to [6–10] and references therein.
Stancu [11, 12] introduced generalizations of Bernstein polynomials with one and two parameters (resp.), satisfying the condition , asfor any . Stancu type generalization of approximation operators present better approach depending on . Therefore, this kind of generalizations and their approximation properties have been studied intensively. We refer the readers to [13–17] and references therein. Mishra et al. [18, 19], V. N. Mishra, and L. N. Mishra  have established very interesting results on approximation properties of various functional classes using different types of positive linear summability operators.
The purpose of this paper is to introduce a new Stancu type generalization of the operators defined in  asBy the definition of operators, it is clear that is positive and linear. For , reduces to operators defined in . In this study we obtain the rate of convergence for the functions having the derivatives of bounded variation. Also, in the end of the paper, we study the simultaneous approximation.
2. Auxiliary Results
In order to prove our main results, we need the following lemmas.
Lemma 1. Let the th order moment be defined aswhere , and then, for , we have the following recurrence relation:Furthermore, is polynomial of degree in and
Proof. By definition of , taking the derivative of , we get Hence, using the identitywe haveWe can write asTo estimate using , we have Next to estimate using the equalitywe havePutting , we getNow integrating by parts, we getProceeding in a similar manner, we obtain the estimate asCombining the equations, we havewhich is the desired result.
Moments for can be easily obtained by using the above recurrence relation.
Remark 2. For sufficiently large , , and , it can be seen from Lemma 1 that where for the convenient notation.
Remark 3. By using Cauchy-Schwarz inequality, it follows from Remark 2 that, for sufficiently large , , and ,
Lemma 4. Let and ; then, for sufficiently large , we have
Proof. We give the proof for only first inequality, and the other is similar. Using Remark 2 with , for sufficiently large and and , we have
Lemma 5. Suppose is times differentiable on such that , for some integer as . Then, for any , and , we have
Proof. Using the identityOne can observe that, even in case , the above identity is true with the condition . Thus, applying (26), we havewhich means that the identity is satisfied for . Let us suppose that the result holds for ; that is,Also, from (26) we can writeand, integrating by parts the last integral, we haveHence we havein which the result is true for , and hence by mathematical induction the proof of the lemma is completed.
3. Main Results
Throughout the paper by we denote the class of absolutely continuous functions on (where is a some positive integer) satisfying the conditions:(i) and ,(ii)the function has the first derivative on the interval which coincide almost everywhere with a function which is of bounded variation on every finite subinterval of . It can be observed that for all functions we can have the representation
Theorem 6. Let , , and . Then, for and sufficiently large , we havewhere is a constant which may be different on each occurrence.
Proof. Using the mean value theorem, we haveAlso, using the identitywhere we haveThus, using the above identities, we can writeAlso, it can be verified that Combining (38)–(40), we getApplying Remark 2 and Lemma 1 in above equation, we haveIn order to complete the proof of the theorem, it suffices to estimate the terms and . Applying Remark 2 with , we get For estimating the integralwe proceed as follows: since implies that so by Schwarz inequality and Lemma 1,By using Hölder’s inequality and Remark 2 (), we get the estimate as follows: Collecting the estimates from (43)–(46), we obtainOn the other hand, to estimate by applying Lemma 4 with and integration by parts, we have