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(p,q)‐Generalization of Szász–Mirakyan operators

In this paper, we introduce new modifications of Szasz–Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p = 1, the previous results for q-Sz asz–Mirakyan operators are captured. Copyright

Asymptotic Formulas for Generalized Szász-Mirakyan Operators

In the present paper, we consider the general Szasz-Mirakyan operators and investigate their asymptotic behaviours. We obtain quantitative Voronovskaya and quantitative Gruss type Voronovskaya theorems using the weighted modulus of continuity. The particular cases are presented for classical Szasz-Mirakyan operators.

Quantitative q-Voronovskaya and q-Grüss–Voronovskaya-type results for q-Szász operators

Abstract In the present paper, we mainly study quantitative Voronovskaya-type theorems for q-Szász operators defined in [20]. We consider weighted spaces of functions and the corresponding weighted modulus of continuity. We obtain the quantitative q-Voronovskaya-type theorem and the q-Grüss–Voronovskaya-type theorem in terms of the weighted modulus of continuity of q-derivatives of the approximated function. In this way, we either obtain the rate of pointwise convergence of q-Szász operators or we present these results for a subspace of continuous functions, although the classical ones are valid for differentiable functions.

On Pointwise Convergence of q-Bernstein Operators and Their q-Derivatives

Pointwise convergence of q-Bernstein polynomials and their q-derivatives in the case of 0 < q < 1 is discussed. We study quantitative Voronovskaya type results for q-Bernstein polynomials and their q-derivatives. These theorems are given in terms of the modulus of continuity of q-derivative of f which is the main interest of this article. It is also shown that our results hold for continuous functions although those are given for two and three times continuously differentiable functions in classical case.

Approximation by bivariate (p,q)-Baskakov–Kantorovich operators

Abstract The present paper deals with the bivariate ( p , q ) {(p,q)} -Baskakov–Kantorovich operators and their approximation properties. First we construct the operators and obtain some auxiliary results such as calculations of moments and central moments, etc. Our main results consist of uniform convergence of the operators via the Korovkin theorem and rate of convergence in terms of modulus of continuity.

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.

On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators

Abstract The present paper deals with a new type of Bernstein-Durrmeyer operators on mobile interval. First, we represent the operators in terms of hypergeometric series. We also establish local and global approximation results for these operators in terms of modulus of continuity. We obtain an asymptotic formula for these operators and in the last section we present better error estimation for the operators using King type approach

Genuine modified Bernstein–Durrmeyer operators

The present paper deals with genuine Bernstein–Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre K

Approximation by Baskakov-Durrmeyer operators based on (p, q)-integers

\mathcal{K}

Approximation by k-th order modifications of Szász—Mirakyan operators

-functional and corresponding modulus of smoothness, quantitative Voronovskaya type theorem and Grüss–Voronovskaya type theorem in quantitative mean are discussed. Finally, the graphic for new operators with special cases and for some values of n is also presented.