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 eigenvectors of the q‐Bernstein operators

In this article, both the eigenvectors and the eigenvalues of the q‐Bernstein operators have been studied. Explicit formulae are presented for the eigenvectors, whose limit behavior is determined both in the case 0 < q < 1 and in the case q > 1. Because the classical case, where q = 1, was investigated exhaustively by S. Cooper and S. Waldron back in 2000, the present article also discusses the related similarities and distinctions with the results in the classical case. Copyright © 2013 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.     

The convergence of q ‐Bernstein polynomials (0 < q < 1) in the complex plane

The paper focuses at the estimates for the rate of convergence of the q ‐Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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。     

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.     

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

In the spectral Petrov‐Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein‐spectral Petrov‐Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.     

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.     

第三型伯恩斯坦插值过程的双结点修正

In this work,the well-known problem put forward by S N Bernstein in 1930 is studied in a deep step.An operator is constructed by revising double interpolation nodes.It is proved that the operator converges to arbitrary continuous functions uniformly and the convergence order is the best.     

关于Bernstein多项式的绝对收敛性

该文研究Ｂｅｒｎｓｔｅｉｎ多项式的绝对收敛性．证明了，对每个ｘ∈［０，１］，一个有界变差函数的Ｂｅｒｎｓｔｅｉｎ多项式序列是绝对｜Ｃ，１｜可和的，而且给出了Ｂｅｒｓｔｅｉｎ多项式序列的绝对｜Ｃ，１｜和式的余项的估计．     

Transformation and reduction formulae for double q‐Clausen hypergeometric series

By means of the Sears transformations, we establish eight general transformation theorems on bivariate basic hypergeometric series. Several transformation, reduction and summation formulae on the double q‐Clausen hypergeometric series are derived as consequences. Copyright © 2007 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.     

Summation formulae for twisted cubic q‐series

A new class of twisted cubic q‐series is investigated by means of the modified Abel lemma on summation by parts. Several remarkable summation and transformation formulae are established for both terminating and nonterminating series.     

关于Bernstein多项式的线性组合

本文对于一类函数建立了Ｂｅｒｎｓｔｅｉｎ多项式线性组合的点态逼近定理。     

Bézier representation of the constrained dual Bernstein polynomials

Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed.     

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     

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.