Abstract:
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

Abstract:
We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators. 1. Introduction The integral modification of Baskakov operators having weight function of some beta basis function are defined as the following: for , , where being the Dirac delta function. The operators defined by (1) were introduced by Gupta [1]; these operators are different from the usual Baskakov-Durrmeyer operators. Actually these operators satisfy condition , where and are constants. In [1], the author estimated some direct results in simultaneous approximation for these operators (1). In particular case , the operators (1) reduce to the operators studied in [2, 3]. In recent years a lot of work has been done on such operators. We refer to some of the important papers on the recent development on similar type of operators [4–9]. The rate of convergence for certain Durrmeyer type operators and the generalizations is one of the important areas of research in recent years. In present article, we extend the studies and here we estimate the rate of convergence for functions having derivative of bounded variation. We denote ; then, in particular, we have By we denote the class of absolutely continuous functions defined on the interval such that,(i) , .(ii)having a derivative on the interval coinciding a.e. with a function which is of bounded variation on every finite subinterval of . It can be observed that all function possess for each a representation 2. Rate of Convergence for Lemma 1 (see [1]). Let the function , , be defined as Then it is easily verified that, for each , , and , and also the following recurrence relation holds: From the recurrence relation, it can be easily be verified that for all , we have . Remark 2. From Lemma 1, using Cauchy-Schwarz inequality, it follows that Lemma 3. Let and be the kernel defined in (1). Then for being sufficiently large, one has ？(a) . ？(b) . Proof. First we prove (a); by using Lemma 1, we have The proof of (b) is similar; we omit the details. Theorem 4. Let , , and . Then for being sufficiently large, we have where the auxiliary function is given by denotes the total variation of on . Proof. By the application of mean value theorem, we have Also, using the identity where we can see that Also, Substitute value of from (12) in (11) and using (14) and (15), we get Using Lemma 1 and Remark 2, we obtain On applying Lemma 3

Abstract:
In this present manuscript, we discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter $\gamma>0$. We compute the moments of these modified operators. Also, establish point-wise convergence, Voronovskaja type asymptotic formula and an error estimation in terms of second order modification of continuity of the function for the operators $B_{n,\gamma}^{\alpha,\beta}(f,x)$.

Abstract:
In this work, we investigate weighted $\alpha$$\beta$-Statistical approximation properties of $q$-Durrmeyer-Stancu operators. Also, give some corrections in limit of $q$-Durrmeyer-Stancu operators defined in \cite{mishra2013short} and discuss their convergence properties.

Abstract:
In the present article, we propose the generalization of Sz\'{a}sz-Mirakyan operators, which is a class of linear positive operators of discrete type depending on a real parameters. We give theorem of degree of approximation, the Voronovskaya Asymptotic formula and statistical convergence.

Abstract:
In the present paper, the authors introduce and investigate new sequences of positive linear operators which include some well known operators as special cases. Here we estimate the rate of convergence for functions having derivatives of bounded variation by families of Jain operators of integral type.

Abstract:
We introduce certain modified Sz\'{a}sz-Mirakyan operators in polynomial weighted spaces of functions of one variable. We studied approximation properties of these operators.

Abstract:
The present paper deals with a generalization of the Baskakov operators. Some direct theorems, asymptotic formula and $A$-statistical convergence are established. Our results are based on a $\rho$ function. These results include the preservation properties of the classical Baskakov operators.

Abstract:
In the manuscript, Voronovskaja type asymptotic formula for function having $q$-derivative of $q$-Durrmeyer operators and $q$-Durrmeyer-Stancu operators are discussed.

Abstract:
The methanolic extract of a blue-green alga and two green algae have been investigated for in vitro antimicrobial activity against Proteus vulgaris, Bacillus cereus, Escherichia coli, Pseudomonas aeruginosa, Aspergillus niger, Aspergillus flavus and Rhizopus nigricans using agar cup-plate method. Blue-green alga, namely, Microchaete tenera ; and green algae, namely, Nitella tenuissima and Sphaeroplea annulina , showed significant antibacterial activity against Pseudomonas aeruginosa . Microchaete tenera showed good antimicrobial activity against Proteus vulgaris and Aspergillus niger. Sphaeroplea annulina showed feeble antifungal activity against Aspergillus flavus .