Harun Karsli

In this paper, we aim to introduce generalized Szász–Mirakyan operators. In this study, we rigorously analyze the convergence behavior of the proposed operators by employing Korovkin’s theorem. The rate of convergence is established through classical analytical frameworks, particularly utilizing the second modulus of continuity and Peetre’s K-functional. Furthermore, an asymptotic formula is derived, and the convergence properties of the operators’ derivatives are thoroughly investigated.

This work is a continuation of the author's very recent studies on the newly introduced wavelet based approximation operators, especially Bernstein operators [14]. The main goal of the present study is to construct and investigate bivariate case of these operators. In accordance with this purpose, we introduce two dimensional wavelet type Bernstein operators via wavelets and examine some characteristic properties together with their approximation results. Moreover, we give some application to bivariate functions and digital image processing.

This article contains construction of a new type of operators using the hypergeometric distribution function, their moments and other approximation properties like direct convergence using the classical Korovkin’s theorem and also the uniform convergence in 𝔏 𝔭 [ 0 , 1 ] space, the rate of convergence using the modulus of continuity, Voronovskaya-type result and we also modified our constructed operators for which we established the statistical convergence property. Further, we have shown the graphical representation of these operators and also shown the numerical comparison between these new operators and some other known operators.

This paper mainly deals with Chlodovsky Bézier variants constructed via compactly supported Daubechies wavelets. We estimate the convergence rate of the aforementioned operators at a fixed point x0 > 0 at which the one-sided limits exist of the target function f. It is evident that the class of operators under consideration encompasses at least the classical version of the Chlodovsky operators along with their B ézier and Kantorovich variants. Therefore, our findings broaden and build upon previous results on Chlodovsky, Chlodovsky Bézier, Chlodovsky-Kantorovich Bézier operators presented in the literature.

In approximation theory, various type Durrmeyer operators have been defined and their approximation properties have been investigated. We include some of them in the article. In this article, we define a Durrmeyer type Neural Network operators within a compact interval [a, b]. Moreover we examine approximation properties of this operators in different function spaces.

In this paper, we present a introduction to Durrmeyer type Lototsky-Chlodowsky operators. We explore their approximation properties within a weighted function space. Subsequently, we derive the rates of convergence utilizing the second modulus of continuity. Moreover, we investigate the convergence rates in the L1 space. In addition, we establish Voronovskaja-type theorem for the considered Durrmeyer type Lototsky-Chlodowsky operators, thereby contributing significant insights into their asymptotic behaviour.

In this study, we define Kantorovich-Stancu type Lototsky-Chlodowsky operators and obtain some approximation properties in weighted space of these operators. Furthermore, we prove Voronovskaja type results for the Kantorovich-Stancu type Lototsky-Chlodowsky operators.

This survey paper provides a historical overview of wavelets and orthonormal systems, alongside recent findings related to linear positive operators reconstructed using wavelets. The first section delves into the historical and chronological development of wavelets, highlighting some of their significant properties. The second section examines linear positive operators constructed through wavelets, discussing their structural characteristics and approximation results. While the list included in this paper is comprehensive, it is not exhaustive. We apologize to any authors whose works on wavelets and wavelet-based operators are not cited in this paper.

In this article, we construct a new sequence of positive linear operators H n : B [ 0 , 1 ] → C [ 0 , 1 ] {H}_{n}:B{[}0,1]\to C{[}0,1] using the hypergeometric distribution of probability theory and the rational values of f at the equally spaced control points k ∕ n k/n ( k = 0 , 1 , … , n ) \left(k=0,1,\ldots ,n) of the unit interval [0,1]. Moreover, we obtain some approximation properties of these operators. It is important to note that hypergeometric distribution has a special interest in probability theory because of its natural behaviour. Namely, unlike all other discrete distributions, the previous steps in the hypergeometric distribution affect the next steps. In other discrete distributions, the process starts from the beginning at each stage, whereas in the hypergeometric distribution, the previous steps determine the structure of the next steps, since the previous steps are not replaced.