In the present paper, we construct a new sequence of BernsteinâKantorovich operators depending on a parameter α . The uniform convergence of the operators and rate of convergence in local âŠ
In the present paper, we construct a new sequence of BernsteinâKantorovich operators depending on a parameter α . The uniform convergence of the operators and rate of convergence in local and global sense in terms of firstâ and secondâorder modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of BernsteinâKantorovich operators and their approximation behaviors.
In this paper, we introduce new modifications of Szasz-Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for âŠ
In this paper, we introduce new modifications of Szasz-Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p=1, the previous results for q-Szasz-Mirakyan operators are captured.
The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $ -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q âŠ
The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $ -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q ) $ -integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.
The present papers deals with the general integral modification of the SzĂĄsz -Mirakyan operators having the weight functions of Baskakov basis functions. Here we estimate the rate of convergence for âŠ
The present papers deals with the general integral modification of the SzĂĄsz -Mirakyan operators having the weight functions of Baskakov basis functions. Here we estimate the rate of convergence for functions having derivatives of bounded variation.
In this article we construct a Durrmeyer modification of the operators introduced by Chen et al. in [10] based on a non-negative real parameter.We establish local approximation, global approximation, Voronovskaja âŠ
In this article we construct a Durrmeyer modification of the operators introduced by Chen et al. in [10] based on a non-negative real parameter.We establish local approximation, global approximation, Voronovskaja type asymptotic theorem.The rate of convergence for differentiable functions whose derivatives are of bounded variation is also obtained.
Pointwise convergence of q-Bernstein polynomials and their q-derivatives in the case of 0 < q < 1 is discussed. We study quantitative Voronovskaya type results for q-Bernstein polynomials and their âŠ
Pointwise convergence of q-Bernstein polynomials and their q-derivatives in the case of 0 < q < 1 is discussed. We study quantitative Voronovskaya type results for q-Bernstein polynomials and their q-derivatives. These theorems are given in terms of the modulus of continuity of q-derivative of f which is the main interest of this article. It is also shown that our results hold for continuous functions although those are given for two and three times continuously differentiable functions in classical case.
The present paper deals with genuine Bernstein-Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre [Formula: see text]-functional and corresponding modulus of âŠ
The present paper deals with genuine Bernstein-Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre [Formula: see text]-functional and corresponding modulus of smoothness, quantitative Voronovskaya type theorem and GrĂŒss-Voronovskaya type theorem in quantitative mean are discussed. Finally, the graphic for new operators with special cases and for some values of n is also presented.
In the present paper, we introduce Bernstein-Chlodowsky-Gadjiev operators taking into consideration the polynomials introduced by Gadjiev and Ghorbanalizadeh [2]. The interval of convergence of the operators is a moved interval âŠ
In the present paper, we introduce Bernstein-Chlodowsky-Gadjiev operators taking into consideration the polynomials introduced by Gadjiev and Ghorbanalizadeh [2]. The interval of convergence of the operators is a moved interval as polynomials given in [2] but grows as n ( â as in the classical Bernstein-Chlodowsky polynomials. Also their knots are shifted and depend on x. We firstly study weighted approximation properties of these operators and show that these operators are more efficient in weighted approximating to function having polynomial growth since these operators contain a factor bn tending to infinity. Secondly we calculate derivative of new Bernstein-Chlodowsky-Gadjiev operators and give a weighted approximation theorem in Lipchitz space for the derivatives of these operators.
The present paper deals with the construction of Baskakov Durrmeyer operators, which preserve the linear functions, in (p,q) -calculus.More precisely, using (p,q) -Gamma function we introduce genuine mixed type Baskakov âŠ
The present paper deals with the construction of Baskakov Durrmeyer operators, which preserve the linear functions, in (p,q) -calculus.More precisely, using (p,q) -Gamma function we introduce genuine mixed type Baskakov Durrmeyer operators having Baskakov and SzĂĄsz basis functions.After construction of the operators and calculations of their moments and central moments, rate of convergence of the operators by means of appropriate modulus of continuity, approximation behaviors for functions belong to Lipschitz class and weighted approximation are explored.
The present paper deals with a new modification of Baskakov operators in which the functions exp( ÎŒ t ) and exp(2 ÎŒ t ), ÎŒ >0 are preserved. Approximation properties âŠ
The present paper deals with a new modification of Baskakov operators in which the functions exp( Ό t ) and exp(2 Ό t ), Ό >0 are preserved. Approximation properties of the operators are captured, ie, uniform convergence and rate of convergence of the operators in terms of modulus of continuity, approximation behaviors of the operators exponential weighted spaces, and pointwise convergence of the operators by means of the Voronovskaya theorem. Advantages of the operators for some special functions are presented.
We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">â</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math>and estimate the rate of convergence for functions having derivatives âŠ
We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant="normal">â</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math>and estimate the rate of convergence for functions having derivatives of bounded variation. Also we present simultenaous approximation by new operators in the end of the paper.
Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. This approximation theorem was extended to more general space of sequences via different way such as statistical âŠ
Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. This approximation theorem was extended to more general space of sequences via different way such as statistical convergence, summation processes. In this work, we introduce a new type of statistical product summability, that is, statistical (C,1) (E,1) summability and further apply our new product summability method to prove Korovkin type theorem. Furthermore, we present a rate of convergence which is uniform in Korovkin type theorem by statistical (C,1) (E,1) summability.
Starting with the wellâ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators âŠ
Starting with the wellâ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bögel space of continuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBSâtype operator are compared.
Very recently,in [4] Chen et.al introduced and considered a new generalization of Bernstein polynomials depending on a patameter α .As classical Bernstein operators, these operators also provide interpolation at the âŠ
Very recently,in [4] Chen et.al introduced and considered a new generalization of Bernstein polynomials depending on a patameter α .As classical Bernstein operators, these operators also provide interpolation at the end points of [0,1] and they have the linear precision property which means those reproduce the linear functions.In this paper we introduce genuine α -Bernstein-Durrmeyer operators.Some approximation results, which include local approximation, error estimation in terms of Ditzian-Totik modulus of smoothness are obtained.Also, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms.
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we âŠ
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we evaluate the rate of convergence by means of suitable moduli of continuity. Throughout some estimates of the rate of convergence, we prove better error estimation than the original operators on certain intervals.
In the present paper, we introduce Stancu type generalization of (p,q)-Szasz-Mirakyan-Baskakov operators and investigate their approximation properties such as weighted approximation, rate of convergence and pointwise convergence.
In the present paper, we introduce Stancu type generalization of (p,q)-Szasz-Mirakyan-Baskakov operators and investigate their approximation properties such as weighted approximation, rate of convergence and pointwise convergence.
In the paper we introduce a general class of linear positive approximation processes defined on bounded and unbounded intervals designed using an appropriate function. Voronovskaya type theorems are given for âŠ
In the paper we introduce a general class of linear positive approximation processes defined on bounded and unbounded intervals designed using an appropriate function. Voronovskaya type theorems are given for these new constructions. Some examples including well known operators are presented.
Abstract The present paper deals with reconstruction of Gamma operators preserving some exponential functions and studies their approximation properties: uniform convergence, rate of convergence, asymptotic formula and saturation. The effectiveness âŠ
Abstract The present paper deals with reconstruction of Gamma operators preserving some exponential functions and studies their approximation properties: uniform convergence, rate of convergence, asymptotic formula and saturation. The effectiveness of new operators compared to classical ones is presented in certain senses as well. The last section is devoted to numerical results which compare the effectiveness of new constructions of Gamma operators.
In 2008 V. Mihesžan constructed a general class of linear positive operators generalizing the Szasz operators. In ÂŽ this article, a Durrmeyer variant of these operators is introduced which is âŠ
In 2008 V. Mihesžan constructed a general class of linear positive operators generalizing the Szasz operators. In Ž this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.
In this study, we introduce newly defined Gamma operators which preserve constants and e 2 ÎŒ · , ÎŒ >0 functions. In accordance with this purpose, we focus on their âŠ
In this study, we introduce newly defined Gamma operators which preserve constants and e 2 Ό · , Ό >0 functions. In accordance with this purpose, we focus on their approximation properties such as uniform convergence, rate of convergence, asymptotic formula, and saturation results. Superior properties of introduced operators have been tested both theoretically and numerically in certain senses to highlight the performance of the new constructions of Gamma operators.
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section âŠ
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section of the function involved. Approximation behaviour of newly constructed operators is investigated at continuity points for log-uniformly continuous functions. The rate of convergence of the series is presented for the same functions by means of logarithmic modulus of continuity. A Voronovskaja type theorem is also presented by means of Mellin derivatives.
Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is âŠ
Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronovskaja-type theorem is obtained.
This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for âŠ
This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for functions that are not necessarily uniformly continuous and bounded on $\mathbb{R}$. In that context we also prove quantitative estimates for the rate of convergence of the family of the above operators in terms of weighted modulus of continuity. Finally, pointwise convergence results in quantitative form by means of Voronovskaja type theorems have been also established.
In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and âŠ
In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation are established. Finally, the theoretical results are demonstrated by using MAPLE software.
This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical âŠ
This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo stretchy="false">[</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend Kingâs approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the SzĂĄsz-Mirakyan operators.
Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted Lp spaces, 1 âŠ
Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted Lp spaces, 1 †p < â for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of A-summability which is a stronger convergence method than ordinary convergence.
The present article deals with local and global approximation behaviors of sampling Durrmeyer operators for functions belonging to weighted spaces of continuous functions. After giving some fundamental notations of sampling âŠ
The present article deals with local and global approximation behaviors of sampling Durrmeyer operators for functions belonging to weighted spaces of continuous functions. After giving some fundamental notations of sampling type approximation methods and presenting well definiteness of the operators on weighted spaces of functions, we examine pointwise and uniform convergence of the family of operators and determine the rate of convergence via weighted modulus of continuity. A quantitative Voronovskaja theorem is also proved in order to obtain rate of pointwise convergence and upper estimate for this convergence. The last section is devoted to some numerical evaluations of sampling Durrmeyer operators with suitable kernels.
In this article, we propose a new sequence of α-Bernstein-Kantorovich type operators based on a parameter ζ, and to study approximation behavior of our operators, we establish local and global âŠ
In this article, we propose a new sequence of α-Bernstein-Kantorovich type operators based on a parameter ζ, and to study approximation behavior of our operators, we establish local and global approximation theorems. Also, we introduce bivariate α-Bernstein-Kantorovich operators and estimate the Voronovskaya type asymptotic theorem and the order of approximation using Peetre's K-functional. Lastly, using Maple software, we show the convergence of our newly defined operators to a certain function by graph.
The present paper deals with construction of newly family of Neural Network operators, that is,Steklov Neural Network operators. By using Steklov type integral, we introduce a new version of Neural âŠ
The present paper deals with construction of newly family of Neural Network operators, that is,Steklov Neural Network operators. By using Steklov type integral, we introduce a new version of Neural Network operators and we obtain some convergence theorems for the family, such as, pointwise and uniform convergence,rate of convergence via moduli of smoothness of order $r$.
In this paper, we introduce Mellin-Steklov exponential samplingoperators of order $r,r\in\mathbb{N}$, by considering appropriate Mellin-Steklov integrals. We investigate the approximation properties of these operators in continuousbounded spaces and $L^p, 1 âŠ
In this paper, we introduce Mellin-Steklov exponential samplingoperators of order $r,r\in\mathbb{N}$, by considering appropriate Mellin-Steklov integrals. We investigate the approximation properties of these operators in continuousbounded spaces and $L^p, 1 \leq p < \infty$ spaces on $\mathbb{R}_+.$ By using the suitablemodulus of smoothness, it is given high order of approximation. Further, we present a quantitative Voronovskaja type theorem and we study the convergence results of newly constructed operators in logarithmic weighted spaces offunctions. Finally, the paper provides some examples of kernels that support the our results.
In the present paper, we analyze the behavior of the exponentialâtype generalized sampling Kantorovich operators when discontinuous signals are considered. We present a proposition for the series , and we âŠ
In the present paper, we analyze the behavior of the exponentialâtype generalized sampling Kantorovich operators when discontinuous signals are considered. We present a proposition for the series , and we prove using this proposition certain approximation theorems for discontinuous functions. Furthermore, we give several examples of kernels satisfying the assumptions of the present theory. Finally, some numerical computations are performed to verify the approximation of discontinuous functions by .
Abstract In the present paper, we introduce a new family of sampling operators, so-called âmodified sampling operatorsâ, by taking a function $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ï</mml:mi> </mml:math> that satisfies the âŠ
Abstract In the present paper, we introduce a new family of sampling operators, so-called âmodified sampling operatorsâ, by taking a function $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ï</mml:mi> </mml:math> that satisfies the suitable conditions, and we study pointwise and uniform convergence of the family of newly introduced operators. We give the rate of convergence of the family of operators via classical modulus of continuity. We also obtain an asymptotic formula in the sense of Voronovskaja. Moreover, we investigate the approximation properties of modified sampling operators in weighted spaces of continuous functions characterized by $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ï</mml:mi> </mml:math> function. Finally, we present examples of some kernels that satisfy the appropriate assumptions. At the end, we present some graphical and numerical representations by comparing the modified sampling operators and the classical sampling operators.
This paper is devoted to an extension of the bivariate generalized Bernstein-Chlodovsky operators preserving the exponential function exp (2,2) where exp (α,ÎČ)=eâαxâÎČy,α,ÎČâR0+ and x,yâ„0. For these operators, we first examine âŠ
This paper is devoted to an extension of the bivariate generalized Bernstein-Chlodovsky operators preserving the exponential function exp (2,2) where exp (α,ÎČ)=eâαxâÎČy,α,ÎČâR0+ and x,yâ„0. For these operators, we first examine the weighted approximation properties for continuous functions in the weighted space, and in the latter case, we also obtain the convergence rate for these operators using a weighted modulus of continuity. Then, we investigate the order of approximation regarding local approximation results via Peetreâs K-functional. We introduce the GBS (Generalized Boolean Sum) operators of generalized Bernstein-Chlodovsky operators, and we estimate the degree of approximation in terms of the Lipschitz class of Bögel continuous functions and the mixed modulus of smoothness. Finally, we provide some numerical and graphical examples with different values to demonstrate the rate of convergence of the constructed operators.
The present article deals with local and global approximation behaviors of sampling Durrmeyer operators for functions belonging to weighted spaces of continuous functions. After giving some fundamental notations of sampling âŠ
The present article deals with local and global approximation behaviors of sampling Durrmeyer operators for functions belonging to weighted spaces of continuous functions. After giving some fundamental notations of sampling type approximation methods and presenting well definiteness of the operators on weighted spaces of functions, we examine pointwise and uniform convergence of the family of operators and determine the rate of convergence via weighted modulus of continuity. A quantitative Voronovskaja theorem is also proved in order to obtain rate of pointwise convergence and upper estimate for this convergence. The last section is devoted to some numerical evaluations of sampling Durrmeyer operators with suitable kernels.
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section âŠ
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section of the function involved. Approximation behaviour of newly constructed operators is investigated at continuity points for log-uniformly continuous functions. The rate of convergence of the series is presented for the same functions by means of logarithmic modulus of continuity. A Voronovskaja type theorem is also presented by means of Mellin derivatives.
Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is âŠ
Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronovskaja-type theorem is obtained.
This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for âŠ
This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for functions that are not necessarily uniformly continuous and bounded on $\mathbb{R}$. In that context we also prove quantitative estimates for the rate of convergence of the family of the above operators in terms of weighted modulus of continuity. Finally, pointwise convergence results in quantitative form by means of Voronovskaja type theorems have been also established.
In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence âŠ
In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of $\log$-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor's expansion of multivariate functions. Using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.
Abstract The present paper deals with reconstruction of Gamma operators preserving some exponential functions and studies their approximation properties: uniform convergence, rate of convergence, asymptotic formula and saturation. The effectiveness âŠ
Abstract The present paper deals with reconstruction of Gamma operators preserving some exponential functions and studies their approximation properties: uniform convergence, rate of convergence, asymptotic formula and saturation. The effectiveness of new operators compared to classical ones is presented in certain senses as well. The last section is devoted to numerical results which compare the effectiveness of new constructions of Gamma operators.
In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and âŠ
In this paper, we present a Durrmeyer type generalization of parametric Bernstein operators. Firstly, we study the approximation behaviour of these operators including a local and global approximation results and the rate of approximation for the Lipschitz type space. The Voronovskaja type asymptotic formula and the rate of convergence of functions with derivatives of bounded variation are established. Finally, the theoretical results are demonstrated by using MAPLE software.
The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present âŠ
The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present some qualitative and quantitative results concerning the approximation by such Boolean sums. The general results are illustrated by examples.
In this study, we introduce newly defined Gamma operators which preserve constants and e 2 ÎŒ · , ÎŒ >0 functions. In accordance with this purpose, we focus on their âŠ
In this study, we introduce newly defined Gamma operators which preserve constants and e 2 Ό · , Ό >0 functions. In accordance with this purpose, we focus on their approximation properties such as uniform convergence, rate of convergence, asymptotic formula, and saturation results. Superior properties of introduced operators have been tested both theoretically and numerically in certain senses to highlight the performance of the new constructions of Gamma operators.
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we âŠ
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we evaluate the rate of convergence by means of suitable moduli of continuity. Throughout some estimates of the rate of convergence, we prove better error estimation than the original operators on certain intervals.
In the paper we introduce a general class of linear positive approximation processes defined on bounded and unbounded intervals designed using an appropriate function. Voronovskaya type theorems are given for âŠ
In the paper we introduce a general class of linear positive approximation processes defined on bounded and unbounded intervals designed using an appropriate function. Voronovskaya type theorems are given for these new constructions. Some examples including well known operators are presented.
This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical âŠ
This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mo stretchy="false">[</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend Kingâs approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the SzĂĄsz-Mirakyan operators.
Starting with the wellâ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators âŠ
Starting with the wellâ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bögel space of continuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBSâtype operator are compared.
The present paper deals with a new modification of Baskakov operators in which the functions exp( ÎŒ t ) and exp(2 ÎŒ t ), ÎŒ >0 are preserved. Approximation properties âŠ
The present paper deals with a new modification of Baskakov operators in which the functions exp( Ό t ) and exp(2 Ό t ), Ό >0 are preserved. Approximation properties of the operators are captured, ie, uniform convergence and rate of convergence of the operators in terms of modulus of continuity, approximation behaviors of the operators exponential weighted spaces, and pointwise convergence of the operators by means of the Voronovskaya theorem. Advantages of the operators for some special functions are presented.
The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine BernsteinâDurrmeyer variants. These new operators depend on a certain âŠ
The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine BernsteinâDurrmeyer variants. These new operators depend on a certain function Ï defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the DitzianâTotik modulus of smoothness are studied. Quantitative Voronovskajaâtype theorems and GrĂŒssâVoronovskajaâtype theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given.
The present paper deals with genuine Bernstein-Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre [Formula: see text]-functional and corresponding modulus of âŠ
The present paper deals with genuine Bernstein-Durrmeyer operators which preserve some certain functions. The rate of convergence of new operators via a Peetre [Formula: see text]-functional and corresponding modulus of smoothness, quantitative Voronovskaya type theorem and GrĂŒss-Voronovskaya type theorem in quantitative mean are discussed. Finally, the graphic for new operators with special cases and for some values of n is also presented.
In this note we construct the BâŹzier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first âŠ
In this note we construct the BâŹzier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In the last section, we study the quantitative Voronovska ja type theorem
In this article we construct a Durrmeyer modification of the operators introduced by Chen et al. in [10] based on a non-negative real parameter.We establish local approximation, global approximation, Voronovskaja âŠ
In this article we construct a Durrmeyer modification of the operators introduced by Chen et al. in [10] based on a non-negative real parameter.We establish local approximation, global approximation, Voronovskaja type asymptotic theorem.The rate of convergence for differentiable functions whose derivatives are of bounded variation is also obtained.
The present paper deals with the construction of Baskakov Durrmeyer operators, which preserve the linear functions, in (p,q) -calculus.More precisely, using (p,q) -Gamma function we introduce genuine mixed type Baskakov âŠ
The present paper deals with the construction of Baskakov Durrmeyer operators, which preserve the linear functions, in (p,q) -calculus.More precisely, using (p,q) -Gamma function we introduce genuine mixed type Baskakov Durrmeyer operators having Baskakov and SzĂĄsz basis functions.After construction of the operators and calculations of their moments and central moments, rate of convergence of the operators by means of appropriate modulus of continuity, approximation behaviors for functions belong to Lipschitz class and weighted approximation are explored.
In 2008 V. Mihesžan constructed a general class of linear positive operators generalizing the Szasz operators. In ÂŽ this article, a Durrmeyer variant of these operators is introduced which is âŠ
In 2008 V. Mihesžan constructed a general class of linear positive operators generalizing the Szasz operators. In Ž this article, a Durrmeyer variant of these operators is introduced which is a method to approximate the Lebesgue integrable functions. First, we derive some indispensable auxiliary results in the second section. We present a quantitative Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.
Very recently,in [4] Chen et.al introduced and considered a new generalization of Bernstein polynomials depending on a patameter α .As classical Bernstein operators, these operators also provide interpolation at the âŠ
Very recently,in [4] Chen et.al introduced and considered a new generalization of Bernstein polynomials depending on a patameter α .As classical Bernstein operators, these operators also provide interpolation at the end points of [0,1] and they have the linear precision property which means those reproduce the linear functions.In this paper we introduce genuine α -Bernstein-Durrmeyer operators.Some approximation results, which include local approximation, error estimation in terms of Ditzian-Totik modulus of smoothness are obtained.Also, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms.
In the present paper, we introduce Stancu type generalization of (p,q)-Szasz-Mirakyan-Baskakov operators and investigate their approximation properties such as weighted approximation, rate of convergence and pointwise convergence.
In the present paper, we introduce Stancu type generalization of (p,q)-Szasz-Mirakyan-Baskakov operators and investigate their approximation properties such as weighted approximation, rate of convergence and pointwise convergence.
Pointwise convergence of q-Bernstein polynomials and their q-derivatives in the case of 0 < q < 1 is discussed. We study quantitative Voronovskaya type results for q-Bernstein polynomials and their âŠ
Pointwise convergence of q-Bernstein polynomials and their q-derivatives in the case of 0 < q < 1 is discussed. We study quantitative Voronovskaya type results for q-Bernstein polynomials and their q-derivatives. These theorems are given in terms of the modulus of continuity of q-derivative of f which is the main interest of this article. It is also shown that our results hold for continuous functions although those are given for two and three times continuously differentiable functions in classical case.
We give some Voronovskaja formula for linear combinations of generalized sampling operators and we furnish also a quantitative version in terms of the classical Peetre K -functional. This provides a âŠ
We give some Voronovskaja formula for linear combinations of generalized sampling operators and we furnish also a quantitative version in terms of the classical Peetre K -functional. This provides a better order of approximation in the asymptotic formula. We apply the general theory to various kernels: Bochner-Riesz kernel, translates of B-splines and Jackson type kernel.
A powerful new method of inversion of the Laplace transform, based on its recently discovered eigenvalues and eigenfunctions, is applied to the problem of inverting light scattering data from a âŠ
A powerful new method of inversion of the Laplace transform, based on its recently discovered eigenvalues and eigenfunctions, is applied to the problem of inverting light scattering data from a polydisperse molecular suspension.
Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is âŠ
Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronovskaja-type theorem is obtained.
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section âŠ
This paper is devoted to construction of multidimensional Kantorovich modifications of exponential sampling series, which allows to approximate suitable measurable functions by considering their mean values on just one section of the function involved. Approximation behaviour of newly constructed operators is investigated at continuity points for log-uniformly continuous functions. The rate of convergence of the series is presented for the same functions by means of logarithmic modulus of continuity. A Voronovskaja type theorem is also presented by means of Mellin derivatives.
Abstract In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous âŠ
Abstract In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.
We build a framework for R(p,q)-deformed calculus, which provides a method of computation for deformed R(p,q)-derivative and integration, generalizing known deformed derivatives and integrations of analytic functions defined on a âŠ
We build a framework for R(p,q)-deformed calculus, which provides a method of computation for deformed R(p,q)-derivative and integration, generalizing known deformed derivatives and integrations of analytic functions defined on a complex disc as particular cases corresponding to conveniently chosen meromorphic functions. Under prescribed conditions, we define the R(p,q)-derivative and integration. Relevant examples are also given.
A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one âŠ
A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maxmum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of "ideal" systems which transmit at this maxmum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.
The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $ -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q âŠ
The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $ -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q ) $ -integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.
In this paper, we introduce new modifications of Szasz-Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for âŠ
In this paper, we introduce new modifications of Szasz-Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p=1, the previous results for q-Szasz-Mirakyan operators are captured.
The author presents a modification of the Baskakov operator for the intervals [0,bn], where bn is an increasing sequence of positive numbers with either finite or infinite limit. Convergence properties âŠ
The author presents a modification of the Baskakov operator for the intervals [0,bn], where bn is an increasing sequence of positive numbers with either finite or infinite limit. Convergence properties of such an operator for continuous and differentiable functions in weighted space are established.
Abstract Investigation of representations of two-parametric quantum groups leads to two-parametric differentiation and integration which are generalizations of the well known q-differentiation and q-integration. In this paper these differentiation and âŠ
Abstract Investigation of representations of two-parametric quantum groups leads to two-parametric differentiation and integration which are generalizations of the well known q-differentiation and q-integration. In this paper these differentiation and integration are studied. Their connection with q-differentiation and q-integration is derived. The p,q-hypergeometric functions are introduced. Their relation to the basic hypergeometric functions is studied. It is emphasized that investigation of some operators of representations of quantum algebras leads to orthogonal polynomials determining their spectral measures and their spectra. Operators of representations of the quantum algebra Upq :(su) and of the p,q-oscillator algebra are studied. Some of these operators lead to unknown orthogonal polynomials. Other ones give particular cases of q-Askey-Wilson polynomials and g-Hermite polynomials with the base r = (pq) 1/2. Keywords: p,q-differentiation p,q-integration p,q-hypergeometric function q-Askey-Wilson polynomials q-Hermite polynomialsquantum groups p,q-oscillator algebraMSC (1991): 17B3733D1533D2533D3533D8039A7047B2581R50
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we âŠ
In this paper we introduce a generalization of Bernstein-Chlodovsky operators that preserves the exponential function $e^{-2x}$ $(x \geq 0)$. We study its approximation properties in several function spaces, and we evaluate the rate of convergence by means of suitable moduli of continuity. Throughout some estimates of the rate of convergence, we prove better error estimation than the original operators on certain intervals.
We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the âŠ
We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]_{p,q} = (p^n - q^n)/(p-q). This twin-basic number occurs in the theory of two-parameter quantum algebras and has been introduced independently in combinatorics. The (p,q)-identities thus derived, with doubling of the number of parameters, offer more choices for manipulations; for example, results that can be obtained via the limiting process of confluence in the usual q-series framework can be obtained by simpler substitutions. The q-results are of course special cases of the (p,q)-results corresponding to choosing p = 1. This also provides a new look for the q-identities.
Let Æ(x) be a given function of a variable x . We shall suppose that Æ(x) is a one-valued analytic function, so that its Taylor's expansion in any part of âŠ
Let Æ(x) be a given function of a variable x . We shall suppose that Æ(x) is a one-valued analytic function, so that its Taylor's expansion in any part of the plane of the complex variable x can be derived from its Taylor's expansion in any other part of the plane by the process of analytic continuation.
In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence âŠ
In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of $\log$-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor's expansion of multivariate functions. Using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.
Here, using Mellin derivatives, a different notion of moment and a suitable modulus of continuity, we state a quantitative Voronovskaja approximation formula for a general class of Mellin convolution operators. âŠ
Here, using Mellin derivatives, a different notion of moment and a suitable modulus of continuity, we state a quantitative Voronovskaja approximation formula for a general class of Mellin convolution operators. This gives a direct approach to the study of pointwise approximation of such operators, without using the Fourier analysis and its results. Various applications to important specific examples are given.
Abstract In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using âŠ
Abstract In this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described.
For pt.I see ibid., vol.7, p.1-20 (1991). The general theory of exponential sampling for the inversion of Mellin-type kernels presented in part I of this paper is applied to some âŠ
For pt.I see ibid., vol.7, p.1-20 (1991). The general theory of exponential sampling for the inversion of Mellin-type kernels presented in part I of this paper is applied to some practical inversion problems encountered in laser scattering experiments to determine particle size.
In this paper, the problem of the order of approximation for the multivariate sampling Kantorovich operators is studied. The cases of uniform approximation for uniformly continuous and bounded functions/signals belonging âŠ
In this paper, the problem of the order of approximation for the multivariate sampling Kantorovich operators is studied. The cases of uniform approximation for uniformly continuous and bounded functions/signals belonging to Lipschitz classes and the case of the modular approximation for functions in Orlicz spaces are considered. In the latter context, Lipschitz classes of Zygmund-type which take into account of the modular functional involved are introduced. Applications to $L^p(\R^n)$, interpolation and exponential spaces can be deduced from the general theory formulated in the setting of Orlicz spaces. The special cases of multivariate sampling Kantorovich operators based on kernels of the product type and constructed by means of Fej\'er's and B-spline kernels have been studied in details.
Recently, Mursaleen et al. (On $(p,q)$ -analogue of Bernstein operators, arXiv:1503.07404 ) introduced and studied the $(p,q)$ -analog of Bernstein operators by using the idea of $(p,q)$ -integers. In this âŠ
Recently, Mursaleen et al. (On $(p,q)$ -analogue of Bernstein operators, arXiv:1503.07404 ) introduced and studied the $(p,q)$ -analog of Bernstein operators by using the idea of $(p,q)$ -integers. In this paper, we generalize the q-Bernstein-Schurer operators using $(p,q)$ -integers and obtain a Korovkin type approximation theorem. Furthermore, we obtain the convergence of the operators by using the modulus of continuity and prove some direct theorems.