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In the current paper a new form of Durrmeyer-type operator involving the beta function is introduced and its approximation properties are studied. We obtain the rate of convergence in different terms. Order of approximation for functions of some special class is also obtained. We establish the Voronovskaja type asymptotic result for this operator and the direct estimate. [ABSTRACT FROM AUTHOR]

This paper deals with the approximation properties of a generalized version of Szász–Mirakjan operators which preserve a x {a^{x}} , a > 1 {a>1} (fixed), and x ≥ 0 {x\geq 0}. The uniform convergence of the operators is studied by using some auxiliary results. Also, error estimations are determined by considering the functions from different spaces. The convergence of the said operators is shown and analyzed by graphics. In the same direction, the proposed operators are compared with Szász–Mirakjan operators for the rate of convergence. A Voronovskaya-type theorem is considered and a comparison with Szász–Mirakjan operators is shown in the sense of convexity. To describe the quantitative means of an asymptotic formula, we quantitatively approach the Voronovskaya-type theorem; moreover, a Grüss–Voronovskaya-type theorem is proved. For further investigations regarding the approximation for functions from various spaces, two significant extensions are added keeping in mind some developments in the L p {L_{p}} -space. One is the Kantorovich variant and the other one is the Durrmeyer modification of the defined operators. Here, the rate of convergence is described by means of the function with a derivative of bounded variation for the Durrmeyer modified operators for which some properties are discussed. Also, A-statistical properties of the Durrmeyer modified operators are established. This paper ends by presenting some significant statements and the conclusion. [ABSTRACT FROM AUTHOR]

Given a function f in the class Lip(α, p) (0 ≤ α 6 1, p ≥ 1), Mittal and Singh (2014) approximated such an f by using trigonometric polynomials, which are the nth terms of either certain Riesz mean or Nörlund mean transforms of the Fourier series representation for f. They showed that the degree of approximation is O((λ(n))-α) and extended two theorems of Leindler (2005) where he had weakened the conditions on {pn} given by Chandra (2002) to more general classes of triangular matrix methods. We obtain the same degree of approximation for a more general class of lower triangular matrices. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain local approximation results for Stancu type generalization of modified Szász-Mirakjan operators. First, we calculate moments of the operators. Some direct results of the operators are investigated. The rate of convergence of the operators is evaluated. In last section of the paper, the Voronovskaya type result is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a Stancu type generalization of modified Srivastava-Gupta operators. We obtain the moments of the operators and then prove the basic convergence theorem. Next, the Voronovskaja type asymptotic formula and some direct results for the above operators are discussed. Also, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Then, we obtain point-wise estimates using the Lipschitz type maximal function and two parameter Lipschitz-type space. Further, we study the A-statistical convergence of these operators. Lastly, we give better estimations of the above operators using King type approach. [ABSTRACT FROM AUTHOR]

In the present paper, we study the mixed summation integral type operators having Baskakov and Szász basis functions in summation and integration form, respectively. We also give here the alternate form of the operators in terms of hypergeometric functions and estimated moments of these operators using hypergeometric series. In the last section, we present some results for operators related to convergence. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the Kantorovich-type Bernstein operators based on q-integers and get a Bohman-Korovkin-type approximation theorem of these operators. We also compute the rate of convergence using the first modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this paper, a kind of modified Baskakov–Schurer–Szász operators (1.2) is introduced. Approximation properties of theses operators are explored: the rate of convergence in terms of the usual moduli of smoothness is given, the convergence in certain weighted spaces is investigated. Furthermore, we study q -analogues of Baskakov–Schurer–Szász operators and its Stancu generalization. Then the better error estimations for the operators (6.3) by using King type approach is investigated. In the last section the Korovkin type weighted statistical approximation property of operators (9.1) is given. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a modification of Szfisz operators involving generalized Brenke type polynomials, First, we investigate Korovkin's type approximation theorem, Next, we established an approximation result for the functions in Lipschitz type space, Furthermore, we discuss the rate of convergence by means of the Ditzian-Totik modulus of smoothness, Lastly, we established an approximation result for the functions having derivatives of bounded variation. [ABSTRACT FROM AUTHOR]

The goal of the present manuscript is to introduce a new sequence of linear positive operators, i.e., Szász-Schurer-Beta type operators to approximate a class of Lebesgue integrable functions. Moreover, we calculate basic estimates and central moments for these sequences of operators. Further, rapidity of convergence and order of approximation are investigated in terms of Korovkin theorem and modulus of smoothess. In subsequent section, local and global approximation properties are studied in various functional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new estimate for the degree of approximation of functions f (x , y) belonging to the generalized Lipschitz class L i p ((ξ 1 , ξ 2) ; r) , r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from L i p ((α , β) ; r) and L i p (α , β) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C , γ , δ) means. [ABSTRACT FROM AUTHOR]

In the present paper, we consider Stancu-Jakimovski-Leviatan-Paltanea operators (SJLPoperators) involving Boas-Buck-Type polynomials. These polynomials include several important cases such as Brenke-type polynomials, Sheffer polynomials and Appell polynomials. We estimate the approximation properties of these operators using modulus of continuity, Peetre's K-functional and functions in Lipschitz type space. Moreover, we also prove some approximation results in the weighted spaces. Lastly, we discuss a Voronoskaja type theorem. [ABSTRACT FROM AUTHOR]

Runge's phenomenon reveals that interpolation methods lack uniform convergence of the sequence of the thus constructed polynomials to the function. The linear positive operators, however, can be relied on to ensure convergence across the whole domain. Further, linear positive operators can be modified suitably to approximate integrable functions. In this paper, we prove theoretical results on the estimates for the rate of approximation of a new sequence of operators. In addition, we find an error estimate for a larger class of functions, the functions of bounded variation. Finally, the theory is supported by suitable numerical examples, and the convergence estimates of the proposed operator are compared with the classical modified Bernstein–Durrmeyer operator. [ABSTRACT FROM AUTHOR]

In this paper, bivariate Sádsz-Mirakjan type operators are introduced along with the estimation of its approximation properties and its rate of convergence. Furthermore, to check the asymptotic behavior of the said bivariate operators, Voronovskaya type theorem is proved and determined the simultaneous approximation of the operators for first order partial derivative. Finally, the convergence behavior of the bivariate operators are obtained through graphical representation and validated the numerical results with the results of Sádsz-Mirakjan operators for the function of two variables. [ABSTRACT FROM AUTHOR]