A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn

Let L_n(f, x)(f ε C)[0, ∞)) be a Bernstein-type approximation operator as defined and studied by Bleimann, Butzer, and Hahn. Probabilistic arguments are used to simplify and sharpen some of their results. The rates of convergence are given in terms of the first and second moduli of continuity. Moreover, an appropriate limit of L_n is shown to be the well-known Szasz operator.     

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.     

Rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators

In the present paper, the authors make use of some results in probability theory with a view to estimating the rate of convergence for the Bézier variant of the Bleimann–Butzer–Hahn operators for functions of bounded variation.     

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.     

Statistical approximation properties of q-Bleimann,Butzer and Hahn operators

The main aim of this study is to introduce a new generalization of q-Bleimann, Butzer and Hahn operators and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical BBH operators. The second aim of this study is to construct a bivariate generalization of the operator and also obtain the statistical approximation properties.     

关于Mohebi-Radjabalipour引理的两个算子

通过两个具体算子的讨论,说明Mohebi-Radjabalipour引理中的条件"ran(λ-T)不是X中的闭集"不能替换成更弱的条件"ran(λ-T)不是Fredholm算子".     

Some Maximal Operators Related to Families of Singular Integral Operators

In this paper, we shall study L^p-boundedness of two kinds of maximal operators related to some families of singular integrals.     

Asymptotic Expansion and Estimate of the Landau Constant

Properties of Landau constant are investigated in this note. A new representation in terms of a hypergeometric function ₃F₂ is given and a property defining the family of asymptotic sequences of Landau constant is formalized. Moreover, we give an other asymptotic expansion of Landau constant by using asymptotic expansion of the ratio of gamma functions in the sense of Poincaré due to Tricomi and Erdélyi.     

Asymptotic Expansion and Estimate of the Landau Constant

Properties of Landau constant are investigated in this note. A new representation in terms of a hypergeometric function ₃F₂ is given and a property defining the family of asymptotic sequences of Landau constant is formalized. Moreover, we give an other asymptotic expansion of Landau constant by using asymptotic expansion of the ratio of gamma functions in the sense of Poincaré due to Tricomi and Erdélyi.     

Axiomatic Potential Theory,Asymptotic Values and Elliptic Differential Operators

In [14] Fernández, Heinonen and Llorente extend the Hornblower's results, about boundary behaviour of subharmonic functions in the unit disc of the complex plane, to subharmonic functions in the unit ball or the upper half space in higher dimensions. In this paper we establish that those results are also valid in the much more general setting of linear axiomatic potential theory. The interest of our general formulation relies on the applications to differential operators. We apply our result to Laplace–Beltrami operator and some uniformly elliptic second order operators in divergence form.     

关于Euler-Mascheroni常数的不等式与渐近展开式（英文）

给出一个递推关系式来确定调和数的渐近展开式的系数.我们建立了Euler-Mascheroni常数的不等式.     

非线性算子的渐近歧点

利用拓扑度方法作为主要工具研究Banach空间中非线性算子的渐近歧点的存在性,但并没有假定非线性算子是渐近线性算子.最后,把一般结果应用于常微分方程的Sturm-Liouville问题.     

On Some Integral Operators Related to the Poisson Equation

In this paper we estimate various norms of some integral operators related to the Poisson equation defined in a bounded domain in the complex plane with vanishing boundary data.     

Asymptotic Expansion of Crocco Solution and the Blasius Constant

We consider the Crocco equation (the reduction of the Blasius equation). The use of this more simple equation for computation of the Blasius constant leads to some unexpected difficulties, which have been unexplained. We computed the asymptotic expansion of the solution to Crocco equation at its singularity. This expansion was unknown before. We describe the structure of the Riemann surface of the Crocco solution at the singularity. These results were used for construction of an effective numerical algorithm, which is based on analytical continuation, for computation of the Blasius constant with an arbitrary and guaranteed accuracy. We computed the Blasius constant with a 100 decimal places.     

On a Constant Related to the Prime Counting Function

Let \({\pi(x)}\) be the number of primes not exceeding x. We produce new explicit bounds for \({\pi(x)}\) and we use them to obtain a fine frame for the remainder term in the asymptotic formula of the sum \({\sum_{2\leq n\leq x}1/\pi(n)}\).     

Overiterated Linear Operators and Asymptotic Behaviour of Semigroups

The results provided hereby are related to the asymptotic behaviour of certain strongly continuous semigroups, which may be expressed in terms of iterates of positive linear operators, in the sense of Altomare’s theory. We present some applications to concrete cases involving continuous and discrete type operators, namely the Beta and the Stancu operators.