Harun Karsli

Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems

Here we give some approximation theorems concerning pointwise convergence for nets of nonlinear integral operators of the form: where the kernel (K λ)λ∈Λ satisfies some general homogeneity assumptions. Here Λ is a nonempty set of indices provided with a topology.

On pointwise convergence of linear integral operators with homogeneous kernels

Here we give some approximation theorems concerning pointwise convergence and rate of pointwise convergence for non-convolution type linear operators of the form: with kernels satisfying some general homogeneity assumptions. Here Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology.

On the rates of convergence of Bernstein–Chlodovsky polynomials and their Bézier-type variants

In this article, we consider the Chlodovsky polynomials C n f and their Bézier variants C n,α f, with α > 0, for locally bounded functions f on the interval [0, ∞). Using the Chanturiya modulus of variation we give estimates for the rates of convergence of C n f (x) and C n,α f (x) at those points x > 0 at which the one-sided limits f (x+), f (x−) exist. The recent results of Karsli and Ibiki [H. Karsli and E. Ibikli, Rate of convergence of Chlodovsky type Bernstein operators for functions of bounded variation, Numer. Funct. Anal. Optim. 28(3–4) (2007), pp. 367–378; H. Karsli and E. Ibikli, Convergence rate of a new Bézier variant of Chlodovsky operators to bounded variation functions, J. Comput. Appl. Math. 212(2) (2008), pp. 431–443.] are essentially improved and extended to more general classes of functions.

Rate of pointwise convergence of a new kind of gamma operators for functions of bounded variation

Abstract In the present paper we investigate the behavior of the operators L n ( f , x ) , defined as L n ( f ; x ) = ( 2 n + 3 ) ! x n + 3 n ! ( n + 2 ) ! ∫ 0 ∞ t n ( x + t ) 2 n + 4 f ( t ) d t , x > 0 , and give an estimate of the rate of pointwise convergence of these operators on a Lebesgue point of bounded variation function f defined on the interval ( 0 , ∞ ) . We use analysis instead of probability methods to obtain the rate of pointwise convergence. This type of study is different from the earlier studies on such a type of operator.

Some approximation properties by q-Szász-Mirakyan-Baskakov-Stancu operators

In the present paper we propose the Stancu type generalization of q-Szász-Mirakyan-Baskakov operators (see e.g. [12, 6]). We apply q-derivatives, and q-Beta functions to obtain the moments of the q-Szász-Mirakyan-Baskakov-Stancu operators. Here we estimate some direct approximation results for these operators.

Some convergence results for nonlinear singular integral operators

Abstract In this paper, we establish some pointwise convergence results for a family of certain nonlinear singular integral operators Tλf of the form (Tλ f) (x)=∫abKλ(t−x, f(t)) dt,      x∈ (a, b),

General Gamma type operators based on q-integers

(T_\lambda \,f)\,(x) = \int\limits_{\rm{a}}^{\rm{b}} {K_\lambda (t - x,\,f(t))\,dt,\,\,\,\,\,\,x \in \,(a,\,b),}

ON THE RATES OF CONVERGENCE OF CHLODOVSKY{DURRMEYER OPERATORS AND THEIR B ¶ EZIER VARIANT

acting on functions with bounded (Jordan) variation on an interval [a, b], as λ → λ0. Here, the kernels 𝕂 = {Kλ}λ∈Λ satisfy some suitable singularity assumptions. We remark that the present study is a continuation and extension of the study of pointwise approximation of the family of nonlinear singular integral operators (1) begun in [18].