In order to approximate integrable functions on the interval [0, 1], Kantorovich gave modified Bernstein polynomials. Later in the year 1967 Durrmeyer [58] considered a more general integral modification of the classical Bernstein polynomials, which were studied first by Derriennic [47]. Also some other generalizations of the Bernstein polynomials are available in the literature. The other most popular generalization as considered by Goodman and Sharma [82], namely, genuine Bernstein-Durrmeyer operators. In this chapter we discuss the q analogues of various integral modifications of Bernstein polynomials. The results were discussed in recent papers [45, 62, 86, 89, 92, 94, 121], etc. q-Bernstein-Kantorovich OperatorsRecently, Dalmanoglu [45] proposed the q-Kantorovich-Bernstein operators asIn case q = 1, the operators (4.1) reduce to well-known Bernstein-Kantorovich operators Direct ResultsFor the operators (4.1), Dalmanoglu [45] obtained the following theorems:Theorem 4.1. If the sequence (q n ) satisfies the conditions lim n→∞ q n = 1 and lim n→∞Proof. First, we haveAlso by definition of q-integral q-Bernstein-Kantorovich Operators 115Again by definition of q-integralTo estimate K n,q (t 2 , x), we haveTherefore using [k + 1] q = q[k] q + 1 and using the similar methods as above, we haveReplacing q by a sequence {q n } such that lim n→∞ q n = 1, it is easily seen that K n,q (t i , x), i = 0, 1, 2 converges uniformly to t i . Thus the result follows by Korovkin's theorem. for all f ∈ C[0, a] and δ n = K n,q ((t − x) 2 , x).

No abstract

The concern of this paper is to introduce a Kantorovich modification of (p, q)-Baskakov operators and investigate their approximation behaviors. We first define a new (p, q)-integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed. MSC: Primary 41A25; secondary 41A36

In the present paper, we introduce Kantorovich modifications of (p, q)-Bernstein operators for bivariate functions using a new (p, q)-integral. We first estimate the moments and central moments. We give the uniform convergence of new operators, rate of convergence in terms of modulus of continuity. The approximations behaviours of the operators for functions having continuous partial derivatives and for functions belong to Lipschitz class are investigated as well.

ABSTRACT. In the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.

We give a new generalization of Bleimann, Butzer, and Hahn operators, which includes qintegers. We investigate uniform approximation of these new operators on some subspace of bounded and continuous functions. In Section 3, we show that the rates of convergence of the new operators in uniform norm are better than the classical ones. We also obtain a pointwise estimation in a general Lipschitz-type maximal function space. Finally, we de?fine a generalization of these new operators and study the uniform convergence of them.Copyright © 2007 A. Aral and O. Dogru. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. IntroductionRecently, in 1997, Phillips [1] used the q-integers in approximation theory where it is considered q-based generalization of classical Bernstein polynomials. It was obtained by replacing the binomial expansion with the general one, the q-binomial expansion. Phillips has obtained the rate of convergence and Voronovskaja-type asymptotic formulae for these new Bernstein operators based on q-integers. Later, some results are established in due course by Phillips et al. (see [2,3,1]). In [4], Barbasu gave Stancu-type generalization of these operators and II'inskii and Ostrovska [5] studied their different convergence properties. Also some results on the statistical and ordinary approximation of functions by Meyer-König and Zeller operators based on q-integers can be found in [6,7], respectively.In [8], Bleimann, Butzer, and Hahn introduced the following operators: 1 + t)) ν for ν = 0,1,2. Some generalization of the operators (1.1) were given in [11][12][13].In this paper, we derive a q-integers-type modification of BBH operators that we call q-BBH operators and investigate their Korovkin-type approximation properties by using the test function (t/(1 + t)) ν for ν = 0,1,2. Also, we define a space of generalized Lipschitz-

No abstract

No abstract