Stancu‐type generalization of Dunkl analogue of Szász–Kantorovich operators

The purpose of the paper is to introduce Stancu‐type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well‐known Korovkin‐type theorem and weighted Korovkin‐type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and second‐order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation properties of Szász‐Mirakyan‐Kantorovich type operators

In this paper, we introduce and study new type Szász‐Mirakyan‐Kantorovich operators using a technique different from classical one. This allow to analyze the mentioned operators in terms of exponential test functions instead of the usual polynomial type functions. As a first result, we prove Korovkin type approximation theorems through exponential weighted convergence. The rate of convergence of the operators is obtained for exponential weights.     

Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

The purpose of this paper is to introduce a family of q‐Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well‐known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and the second‐order modulus of continuity. Furthermore, we obtain the approximation results for bivariate q‐Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.     

A summation‐integral type modification of Szász–Mirakjan operators

In this paper, we introduced a summation‐integral type modification of Szász–Mirakjan operators. Calculation of moments, density in some space, a direct result and a Voronvskaja‐type result, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation properties for King's type modified q‐Bernstein–Kantorovich operators

In the present research article, we introduce the King's type modification of q‐Bernstein–Kantorovich operators and investigate some approximation properties. We show comparisons and present some illustrative graphics for the convergence of these operators to some function. Copyright © 2015 John Wiley & Sons, Ltd.     

Quantitative estimations of bivariate summation‐integral–type operators

In this paper, we study the approximation properties of bivariate summation‐integral–type operators with two parameters . The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0],∞)(×[0],∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation‐integral–type operators and Szász‐Mirakjan‐Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász‐Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first‐order partial derivative of the function.     

Local and global results for modified Szász–Mirakjan operators

In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.     

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

In this paper, we introduce new modifications of Szász–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 ász–Mirakyan operators are captured. Copyright © 2015 John Wiley & Sons, Ltd.     

Korovkin type theorem for non‐tensor Balázs type Bleimann,Butzer and Hahn operators

In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

Szász–Mirakjan type operators providing a better error estimation

In this work, giving a modification of the well-known Szasz–Mirakjan operators, we prove that the error estimation of our operators is better than that of the classical Szasz–Mirakjan operators. Furthermore, we obtain a Voronovskaya type theorem for these modified operators.     

Voronovskaya‐type theorems for Urysohn type nonlinear Bernstein operators

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators.     

Approximation properties of a generalization of positive linear operators

In the present paper we introduce a generalization of positive linear operators and obtain its Korovkin type approximation properties. The rates of convergence of this generalization is also obtained by means of modulus of continuity and Lipschitz type maximal functions. The second purpose of this paper is to obtain weighted approximation properties for the generalization of positive linear operators defined in this paper. Also we obtain a differential equation so that the second moment of our operators is a particular solution of it. Lastly, some Voronovskaja type asymptotic formulas are obtained for Meyer-König and Zeller type and Bleimann, Butzer and Hahn type operators.     

Statistical approximation properties of the Durrmeyer type q‐Bleimann,Butzer, and Hahn operators

In this paper, we introduce a Durrmeyer‐type generalization of q‐Bleimann, Butzer, and Hahn operators based on q‐integers and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. We also compute rates of statistical convergence of these q‐type operators by means of the modulus of continuity and Lipschitz‐type maximal function, respectively. Copyright © 2011 John Wiley & Sons, Ltd.     

q‐Voronovskaya type theorems for q‐Baskakov operators

In the present paper, we prove quantitative q‐Voronovskaya type theorems for q‐Baskakov operators in terms of weighted modulus of continuity. We also present a new form of Voronovskaya theorem, that is, q‐Grüss‐Voronovskaya type theorem for q‐Baskakov operators in quantitative mean. Hence, we describe the rate of convergence and upper bound for the error of approximation, simultaneously. Our results are valid for the subspace of continuous functions although classical ones is valid for differentiable functions. Copyright © 2015 John Wiley & Sons, Ltd.     

Hyponormality of bounded‐type Toeplitz operators

In this paper we deal with the hyponormality of Toeplitz operators with matrix‐valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded‐type Toeplitz operators (i.e., the symbol is a matrix‐valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co‐analytic part of the symbol Φ is a left divisor of the analytic part.     

Approximation numbers and Kolmogorov widths of Hardy‐type operators in a non‐homogeneous case

Let I = [a , b ] ? ?, let 1 < q ≤ p < ∞, let u and v be positive functions with u ∈ L _{p ′} (I ) and v ∈ L _q (I ), and let T : L _p (I ) → L _q (I ) be the Hardy‐type operator given by Given any n ∈ ?, let s _n stand for either the n ‐th approximation number of T or the n ‐th Kolmogorov width of T . We show that where c _pq is an explicit constant depending only on p and q . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A (1 + ϵ)‐approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász Local Lemma*

In his seminal result, Beck gave the first algorithmic version of the Lovász Local Lemma by giving polynomial time algorithms for 2‐coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2‐coloring nonuniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a nonuniform hypergraph in which every edge e_i intersects at most |e_i|2^αk other edges of size at most k, for some small constant α, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

On the asymptotic behaviour of linear combinations of Mellin‐Picard type operators

Here we give a Voronovskaja formula for linear combination of Mellin‐Picard type convolution operators where is the Mellin‐Picard kernel. This approach provides a better order of pointwise approximation.     

A generalization of Turán's theorem

In this paper, we obtain an asymptotic generalization of Turán's theorem. We prove that if all the non‐trivial eigenvalues of a d‐regular graph G on n vertices are sufficiently small, then the largest K_t‐free subgraph of G contains approximately (t ? 2)/(t ? 1)‐fraction of its edges. Turán's theorem corresponds to the case d = n ? 1. © 2005 Wiley Periodicals, Inc. J Graph Theory