Some Approximation Properties of Modified Jain-Beta Operators

Generalization of Szász-Mirakyan operators has been considered by Jain, 1972. Using these generalized operators, we introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function. Approximation properties, the rate of convergence, weighted approximation theorem, and better approximation are investigated for these new operators. At the end, we generalize Jain-Beta operator with three parameters , , and and discuss Voronovskaja asymptotic formula. 1. Introduction For ,？？ , let then Equation (1) is a Poisson-type distribution which has been considered by Consul and Jain [1]. In 1970, Jain [2] introduced and studied the following class of positive linear operators: where and has been defined in (1). The parameter may depend on the natural number . It is easy to see that ; (3) reduces to the well-known Szász-Mirakyan operators [3]. Different generalization of Szász-Mirakyan operator and its approximation properties is studied in [4, 5]. Kantorovich-type extension of was given in [6]. Integral version of Jain operators using Beta basis function is introduced by Tarabie [7], which is as follows: In Gupta et al. [8] they considered integral modification of the Szász-Mirakyan operators by considering the weight function of Beta basis functions. Recently, Dubey and Jain [9] considered a parameter in the definition of [8]. Motivated by such types of operators we introduce new sequence of linear operators as follows: For and , where is defined in (1) and The operators defined by (5) are the integral modification of the Jain operators having weight function of some Beta basis function. As special case, the operators (5) reduced to the operators recently studied in [7]. Also, if and , then the operators (5) turn into the operators studied in [8]. In the present paper, we introduce the operators (5) and estimate moments for these operators. Also, we study local approximation theorem, rate of convergence, weighted approximation theorem, and better approximation for the operators . At the end, we propose Stancu-type generalization of (5) and discuss some local approximation properties and asymptotic formula for Stancu-type generalization of Jain-Beta operators. 2. Basic Results Lemma 1 (see [2]). For ,？？ , one has Lemma 2. The operators , defined by (5) satisfy the following relations: Proof. By simple computation, we get Lemma 3. For , , and with , one has (i) ,(ii) ？ . Lemma 4. For , , one has Proof. Since , , and , we have which is required. 3. Some Local Approximation