Search

Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov–Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian–Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed. [ABSTRACT FROM AUTHOR]

The present paper deals with the approximation properties of the univariate operators which are the generalization of the Kantorovich-Szász type operators involving Brenke type polynomials. We investigate the order of convergence by using Peetre's K-functional and the Ditzian-Totik modulus of smoothness and study the degree of approximation of the univariate operators for continuous functions in a Lipschitz space, a Lipschitz type maximal function and a weighted space. The rate of approximation of functions having derivatives equivalent with a function of bounded variation is also obtained. [ABSTRACT FROM AUTHOR]

The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation. [ABSTRACT FROM AUTHOR]

In this paper, we construct a sequence of modified Stancu-Baskakov operators for a real valued function bounded on $[0,\infty)$ , based on a function $\tau(x)$ . This function $\tau(x)$ is infinite times continuously differentiable on $[0,\infty)$ and satisfy the conditions $\tau (0)=0,~\tau ^{\prime}(x)>0$ and $\tau^{\prime\prime}(x)$ is bounded for all $x\in {}[0,\infty)$ . We study the degree of approximation of these operators by means of the Peetre K-functional and the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja-type theorems are also established in terms of the first order Ditzian-Totik modulus of smoothness. [ABSTRACT FROM AUTHOR]

Recently, Sofonea (Gen. Math. 16:47-54, 2008) considered some relations in the context of quantum calculus associated with the q-derivative operator $D_{q}$ and divided difference. As applications of the post-quantum calculus known as the $(p,q)$ -calculus, we derive several relations involving the $(p,q)$ -derivative operator and divided differences. [ABSTRACT FROM AUTHOR]

In the present paper, firstly we find a number of poles of generating functions of Bernoulli numbers and associated Euler numbers, denoted by n(a,B) and n(a,E), respectively. Secondly, we derive the mean value of a positive logarithm of generating functions of Bernoulli numbers and associated Euler numbers shown as m(2π,B) and m(π,E), respectively. From these properties, we find Nevanlinna characteristic functions which we stated in the paper. Finally, as an application, we show that the generating function of Bernoulli numbers is a normal function. [ABSTRACT FROM AUTHOR]