Ditzian Totik Research Articles

Approximation by Stancu variant of [formula omitted]-Bernstein shifted knots operators associated by Bézier basis function

The current paper presents the λ-Bernstein operators through the use of newly developed variant of Stancu-type shifted knots polynomials associated by Bézier basis functions. Initially, we design the proposed Stancu generated λ-Bernstein operators by means of Bézier basis functions then investigate the local and global approximation results by using the Ditzian–Totik uniform modulus of smoothness of step weight function. Finally we establish convergence theorem for Lipschitz generated maximal continuous functions and obtain some direct theorems of Peetre’s K-functional. In addition, we establish a quantitative Voronovskaja-type approximation theorem.

Approximation by modified Bernstein polynomials based on real parameters

In this paper, we introduce a modified Bernstein-type operators based on two real parameters and study its various approximation properties. We derive some direct results e.g. Voronovkaja type asymptotic theorem, an estimate of error in ordinary as well as in Ditzian Totik modulus of smoothness and an error estimate for functions belonging to the Lipschitz type space. Further, we examine the rate of approximation for a Kirov and Popova type generalization of these operators.

Estimates in direct inequalities for the Szász–Mirakyan operator

This paper deals with the approximation of continuous functions by the classical Szász–Mirakyan operator. We give new bounds for the constant in front of the second order Ditzian–Totik modulus of smoothness in direct inequalities. Asymptotic and non asymptotic results are stated. We use both analytical and probabilistic methods, the latter involving the representation of the operators in terms of the standard Poisson process. A smoothing technique based on a modification of the Steklov means is also applied.

Approximation by α$$ \alpha $$‐Bernstein–Schurer operators and shape preserving properties via q$$ q $$‐analogue

Our work in this article is to construct the ‐ Bernstein–Schurer operators which includes the ‐integers. For these new operators, we discuss the shape preserving properties, namely, monotonicity and convexity. Next, we study the uniformly global approximation in terms of the Ditzian–Totik modulus of continuity and calculate the local direct estimate by Lipschitz‐maximal functions. In the last Voronovskaja‐type approximation theorems are also presented.

Asymptotic and Non-asymptotic Results in the Approximation by Bernstein Polynomials

This paper deals with the approximation of functions by the classical Bernstein polynomials in terms of the Ditzian–Totik modulus of smoothness. Asymptotic and non-asymptotic results are respectively stated for continuous and twice continuously differentiable functions. By using a probabilistic approach, known results are either completed or strengthened.

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative Voronovskaja-type theorem for our newly constructed operators. Moreover, we investigate the approximation of functions with derivatives of bounded variation (DBV) of the aforesaid operators.

Approximation Properties of Some Modified Szász–Mirakjan–Kantorovich Operators

The present article deals with approximation results obtained by means of the Lipschitz maximal function, the Ditzian–Totik modulus of smoothness, and a Lipschitz-type space having two parameters for the summation-integral type operators defined by Mishra and Yadav [22]. Further, we determine the rate of convergence in terms of a derivative of bounded variation. To estimate the quantitative results of the defined operators, we establish quantitative Voronovskaya-type and Grüss-type theorems. Moreover, examples are given with graphical representation to support the main results.

Modified Bernstein–Durrmeyer Type Operators

We constructed a summation–integral type operator based on the latest research in the linear positive operators area. We establish some approximation properties for this new operator. We highlight the qualitative part of the presented operator; we studied uniform convergence, a Voronovskaja-type theorem, and a Grüss–Voronovskaja type result. Our subsequent study focuses on a direct approximation theorem using the Ditzian–Totik modulus of smoothness and the order of approximation for functions belonging to the Lipschitz-type space. For a complete image on the quantitative estimations, we included the convergence rate for differential functions, whose derivatives were of bounded variations. In the last section of the article, we present two graphs illustrating the operator convergence.

Optimal stochastic Bernstein polynomials in Ditzian–Totik type modulus of smoothness

We introduce a family of symmetric stochastic Bernstein polynomials based on order statistics, and establish the order of convergence in probability in terms of the second order Ditzian–Totik type modulus of smoothness on the interval [0,1], which epitomizes an optimal pointwise error estimate for the classical Bernstein polynomial approximation. Monte Carlo simulation results (presented at the end of the article) show that this new approximation scheme is efficient and robust.

Bézier variant of Bernstein–Durrmeyer blending-type operators

In this paper, we construct the Bézier variant of the Bernstein–Durrmeyer-type operators. First, we estimated the moments for these operators. In the next section, we found the rate of approximation of operators [Formula: see text] using the Lipschitz-type function and in terms of Ditzian–Totik modulus of continuity. The rate of convergence for functions having derivatives of bounded variation is discussed. Finally, the graphical representation of the theoretical results and the effectiveness of the defined operators are given.

Gamma Generalization Operators Involving Analytic Functions

In the present paper, we give an operator with the help of a generalization of Boas–Buck type polynomials by means of Gamma function. We have approximation properties and moments. The rate of convergence is given by the Ditzian–Totik first order modulus of smoothness and the K-functional. Furthermore, we obtain the point-wise estimations for this operator.

Moduli of Smoothness for Weighted Lp ,0 < p < 1 Approximation

Moduli of smoothness are intended for mathematicians working in approximation theory, numerical analysis and real analysis. Measuring the smoothness of a function by differentiability is too crude for many purposes in approximation theory. For many purposes many authors defined many types of moduli of smoothness. For easier work and for weighted approximation, we introduce some types of moduli of smoothness. In this paper we introduce a new moduli of smoothness for functions ƒ Lp [-1,1] n Cr_1 (-1,1), 0 < p < 1, r ≥ 1, that have an (r — 1)st locally bsolutely cotinous derivative in (-1,1), and such that φrfris in Lp [-1,1], where φ(x)= (1 - x 2)1/2. These moduli of smoothness are used to describe the smoothness of the derivative of functions in Lp spaces for 0 < p < 1, also it is equivalent to specific “weighted Ditzian-Totik (DT) moduli” of smoothness, but our definition is more straightforward and easier. we also introduce some relate properties with some relations and we prove the equivalence to “K- functional”

Approximation properties of generalized Baskakov operators

The present article is a continuation of the work done by Aral and Erbay [ 1 ]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bogel continuous and Bogel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.

Approximation of functions by a new class of generalized Bernstein–Schurer operators

In this work, we construct a new kind of Bernstein–Schurer operators which includes non-negative real parameter $$\alpha $$ . We study some shape preserving properties, namely, monotonicity and convexity of the new operators. We obtain global approximation formula in terms of Ditzian–Totik uniform modulus of smoothness of first and second order and calculate the local direct estimate of the rate of convergence with the help of Lipschitz-type function for our operators. The Voronovskaja-type approximation theorems of the new operators are presented. Finally, in the last section, we provide some graphs, MATLAB code and numerical examples in order to illustrate the significance of our newly constructed operators.

Bézier–Bernstein–Durrmeyer type operators

In this note, we construct the Bezier variant of the Bernstein–Durrmeyer type operators. We present local results, a direct approximation theorem by using the Ditzian–Totik modulus of smoothness and a quantitative Voronovskaja type theorem with the help of the Ditzian–Totik modulus of continuity. The rate of convergence for differential functions whose derivatives are of bounded variation is also established. Finally, we show that the numerical examples which illustrate the authenticity of the theoretical results and the effectiveness of the defined operators.

Generalized Bernstein–Durrmeyer operators of blending type

In this article, we present the Durrmeyer variant of generalized Bernstein operators that preserve the constant functions involving a non-negative parameter $$\rho $$ . We derive the approximation behaviour of these operators including a global approximation theorem via Ditzian–Totik modulus of continuity and the order of convergence for the Lipschitz type space. Furthermore, we study a Voronovskaja type asymptotic formula, local approximation theorem by means of second order modulus of smoothness and the rate of approximation for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Lastly, we illustrate the convergence of these operators for certain functions using Maple software.

Further results concerning some general Durrmeyer type operators

The present paper deals with the approximation properties of the Mihesan-Durrmeyer type operators due to Kajla (Adv Oper Theory 2:162–178, 2017) which include well-known operators like Phillips operators (Ann Math 59:325–356, 1954). and Pǎltǎnea operators (Carpathian J Math 24:378–385, 2008) as particular cases. We establish the degree of approximation of the operators in a weighted space and then extend the study to the approximation in a weighted space and the order of convergence of the operators by means of the Ditzian–Totik modulus of smoothness, quantitative and Gruss Voronovskaya type approximation. We construct a variant of our operator that preserves the test functions $$e_0$$ and $$e_2$$ and show that the new operators present a better rate of convergence than the original ones. Next we define an operator which preserves the functions $$e_0$$ and $$e^{-x}$$ and show by numerical examples and illustrations that at least one of the two newly constructed operators presents a better rate of convergence than the original operator.

Convergence Properties of Certain Positive Linear Operators

The present paper deals with the modified positive linear operators that present a better degree of approximation than the original ones. This new construction of operators depend on a certain function $$\varrho $$ defined on [0, 1]. Some approximation properties of these operators are given. Using the first order Ditzian–Totik modulus of smoothness, some Voronovskaja type theorems in quantitative mean are proved. The main results proved in this paper are applied for Bernstein operators, Lupas operators and genuine Bernstein–Durrmeyer operators. By numerical examples we show that depending on the choice of the function $$\varrho $$ , the modified operator presents a better order of approximation than the classical ones.

Quantitative Voronovskaja and Grüss Voronovskaja-Type Theorems for Operators of Kantorovich Type Involving Multiple Appell Polynomials

The purpose of the present paper is to obtain the quantitative Voronovskaja and Gruss Voronovskaja-type theorems by calculating the sixth-order central moment for the Jakimovski–Leviatan operators of Kantorovich type based on multiple Appell polynomials. Also, we study the rate of approximation for functions in a Lipschitz-type space and in terms of the second-order modulus of continuity and Ditzian–Totik modulus of smoothness for continuous functions in the semi-positive axis.

Approximation properties of \u03bb-Kantorovich operators

In the present paper, we study a new type of Bernstein operators depending on the parameter lambdain[-1,1]. The Kantorovich modification of these sequences of linear positive operators will be considered. A quantitative Voronovskaja type theorem by means of Ditzian–Totik modulus of smoothness is proved. Also, a Grüss–Voronovskaja type theorem for λ-Kantorovich operators is provided. Some numerical examples which show the relevance of the results are given.