Higher order limit q‐Bernstein operators

In this paper we give the estimates of the central moments for the limit q‐Bernstein operators. We introduce the higher order generalization of the limit q‐Bernstein operators and using the moment estimations study the approximation properties of these newly defined operators. It is shown that the higher order limit q‐Bernstein operators faster than the q‐Bernstein operators for the smooth functions defined on [0, 1]. Copyright © 2011 John Wiley & Sons, Ltd.     

On the image of the limit q‐Bernstein operator

The limit q‐Bernstein operator B_q emerges naturally as an analogue to the Szász–Mirakyan operator related to the Euler distribution. Alternatively, B_q comes out as a limit for a sequence of q‐Bernstein polynomials in the case 0<q<1. Lately, different properties of the limit q‐Bernstein operator and its iterates have been studied by a number of authors. In particular, it has been shown that B_q is a positive shape‐preserving linear operator on C[0, 1] with ∥B_q∥=1, which possesses the following remarkable property: in general, it improves the analytic properties of a function. In this paper, new results on the properties of the image of B_q are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 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.     

Some approximation properties of a Durrmeyer variant of q‐Bernstein–Schurer operators

In this paper, we will propose a Durrmeyer variant of q‐Bernstein–Schurer operators. A Bohman–Korovkin‐type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

Bernstein operators for extended Chebyshev systems

Let U_n ⊂ Cⁿ[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f₀ ∈ U_n is strictly positive and f₁ ∈ U_n has the property that f₁/f₀ is strictly increasing. We search for conditions ensuring the existence of points t₀, …, t_n ∈ [a, b] and positive coefficients α₀, …, α_n such that for all f ∈ C[a, b], the operator B_n:C[a, b] → U_n defined by satisfies B_nf₀ = f₀ and B_nf₁ = f₁. Here it is assumed that p_n,k, k = 0, …, n, is a Bernstein basis, defined by the property that each p_n,k has a zero of order k at a and a zero of order n − k at b.     

One‐dimensional q‐Dirac equation

In this paper, we introduce a q‐analog of 1‐dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self‐adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of . Finally, we give some examples.     

q‐pseudoconvex and q‐holomorphically convex domains

In this article we prove a global result in the spirit of Basener's theorem regarding the relation between q‐pseudoconvexity and q‐holomorphic convexity: we prove that any open subset $\mathrm{\Omega }?{\mathbb{C}}^n$ with smooth boundary, strictly q‐pseudoconvex, is $(q+1)$‐holomorphically convex; moreover, assuming that Ω verifies an additional assumption, we prove that it is q‐holomorphically convex. We also prove that any open subset of ${\mathbb{C}}^n$ is n‐holomorphically convex.     

On an extremal relation of Bernstein operators

In this note we present a new characterization of Bernstein operators by showing that they are the only solution of a certain extremal relation.     

关于Bernstein算子的收敛速度

本文对一类函数建立了Bernstein算子的一致逼近定理,而且给出了其逆定理的一个简短证明。     

Alternative correction equations in the Jacobi‐Davidson method

The correction equation in the Jacobi‐Davidson method is effective in a subspace orthogonal to the current eigenvector approximation, whereas for the continuation of the process only vectors orthogonal to the search subspace are of importance. Such a vector is obtained by orthogonalizing the (approximate) solution of the correction equation against the search subspace. As an alternative, a variant of the correction equation can be formulated that is restricted to the subspace orthogonal to the current search subspace. In this paper, we discuss the effectiveness of this variant. Our investigation is also motivated by the fact that the restricted correction equation can be used for avoiding stagnation in the case of defective eigenvalues. Moreover, this equation plays a key role in the inexact TRQ method [18]. Copyright © 1999 John Wiley & Sons, Ltd.     

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‐Analogs of Packing Designs

A q‐packing design is a selection of k‐dimensional subspaces of such that each t‐dimensional subspace is contained in at most one element of the collection. A successful approach adopted from the Kramer–Mesner method of prescribing a group of automorphisms was applied by Kohnert and Kurz to construct some constant dimension codes with moderate parameters that arise by q‐packing designs. In this paper, we recall this approach and give a version of the Kramer–Mesner method breaking the condition that the whole q‐packing design must admit the prescribed group of automorphisms. Afterwards, we describe the basic idea of an algorithm to tackle the integer linear optimization problems representing the q‐packing design construction by means of a metaheuristic approach. Finally, we give some improvements on the size of q‐packing designs.     

Statistically q‐deformed and tau‐deformed systems

Two classes of statistically deformed systems are known in literature. They are, respectively, the q‐deformed systems (Lavagno and Narayana Swamy, Phys Rev E 2002, 65, 036101) and the κ‐deformed systems (Kaniadakis and Scarfone, Physica A 2002, 305, 69). In this article, a new class, i.e., the tau‐deformed systems, is introduced. For each of these systems, a consistent thermodynamics may be developed. A summary of the main similarities between the thermodynamic properties of q‐deformed and tau‐deformed systems is presented. The deformation outlined in this article is radically different from the nonextensive Tsallis statistics, where the structure of the entropy is rather arbitrary deformed via the logarithmic function. In contrast, the theory of tau‐deformed systems is developed on a purely physical basis. However, one finally shows that the tau‐systems may be described by using a new form of deformed logarithmic function. © 2009 Wiley Periodicals, Inc. Complexity, 2010     

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.     

关于Bernstein-Fan算子的导数与连续模

对具有三角形波基函数的 Bernstein- Fan插值算子建立了导数与连续模之间的关系 .     

On simultaneous approximation of the Bernstein Durrmeyer operators

The present paper deals with the study of the rate of convergence of the Bézier variant of certain Bernstein Durrmeyer type operators in simultaneous approximation.     

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.