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Octavian Agratini

Octavian Agratini
Babeş-Bolyai University AND Tiberiu Popoviciu Numerical Institute · Department of Mathematics

Professor

About

100
Publications
3,183
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667
Citations
Additional affiliations
October 2018 - present
Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy
Position
  • Senior Researcher
October 2002 - present
Babeş-Bolyai University Faculty of Mathematics and Computer Science
Position
  • Professor (Full)

Publications

Publications (100)
Article
The starting point is an approximation process consisting of linear and positive operators. The purpose of this note is to establish the limit of the iterates of some multidimensional approximation operators. The main tool is a Perov’s result which represents a generalization of Banach fixed point theorem. In order to support the theoretical aspect...
Article
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The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indica...
Article
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"This note focuses on a sequence of linear positive operators of integral type in the sense of Kantorovich. The construction is based on a class of discrete operators representing a new variant of Jain operators. By our statements, we prove that the integral family turns out to be useful in approximating continuous signals de ned on unbounded inter...
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In this paper, we define and study a general class of convolution operators based on Landau operators. A property of these new operators is that they reproduce the affine functions, a feature less commonly encountered by integral type operators. Approximation properties in different function spaces are obtained, including quantitative Voronovskaya-...
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This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in \(L_p\) spaces and in weighted \(L_p\) spaces \((1\le p<\infty )\). Also, we give an extension of the operators to approximate real-valued vector functions. In...
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In this paper we consider a general class of linear positive processes of integral type. These operators act on functions defined on unbounded interval. Among the particular cases included are Durrmeyer–Jain operators, Păltănea–Szász–Mirakjan operators and operators using Baskakov–Szász type bases. We focus on highlighting some approximation target...
Article
In the present note we modify a linear positive Markov process of discrete type by using so called multiplicative calculus. In this framework, a convergence property and the error of approximation are established. In the final part some numerical examples are delivered.
Article
Starting from a discrete linear approximation process that has the ability to turn polynomials into polynomials of the same degree, we introduce an integral generalization by using Beta-type bases. Some properties of this new sequence of operators are investigated in unweighted and weighted spaces of functions defined on unbounded interval. In our...
Article
This note aims to highlight a general class of discrete linear and positive operators, focusing on some of their approximation properties. The construction includes a sequence of positive numbers (λn), strictly decreasing. Imposing additional conditions on it, we set the convergence of the operators towards the identity operator and establish upper...
Article
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The paper is focused on general sequences of discrete linear operators, say \((L_n)_{n\ge 1}\). The special case of positive operators is also to our attention. Concerning the quantity \({\Delta } (L_n,f,g):=L_n(fg)-(L_n f)(L_n g), f\) and g belonging to some certain spaces, we propose different estimates. Firstly, we study its asymptotic behavior...
Article
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On the last five decades the interest of the study of positive approximation processes have emerged with growing evidence. A special place is occupied by the in-depth study of classical operators. The most eloquent example is Bernstein operator which represents a permanent challenge for the researches in the mentioned field. However, in this synthe...
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This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose b...
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We derive the complete asymptotic expansion for the quasi-interpolants of Gauß–Weierstraß operators \(W_{n}\) and their left quasi-interpolants \(W_{n}^{\left[ r\right] }\) with explicit representation of the coefficients. The results apply to all locally integrable real functions f on \(\mathbb {R}\) satisfying the growth condition \({f}\left( t\r...
Chapter
Sequences of binomial operators introduced by using umbral calculus are investigated from the point of view of statistical convergence. This approach is based on a detailed presentation of delta operators and their associated basic polynomials. Bernstein–Sheffer linear positive operators are analyzed, and some particular cases are highlighted: Chen...
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The paper aims to study two classes of linear positive operators representing modifications of Picard and Gauss operators. The new operators reproduce both constants and a given exponential function. Approximation properties in polynomial weighted spaces are investigated and the speed of convergence is measured using a certain weighted modulus of s...
Article
Starting from positive linear operators which have the capability to reproduce affine functions, we design integral operators of Kantorovichtype which enjoy by the same property. We focus to show that the error of approximation can be smaller than in classical Kantorovich construction on some subintervals of its domain. Special cases are presented.
Article
In this note we spotlight the linear and positive operators of discrete type \({{{(R_n)}_{n\geqq1}}}\) known as Balázs–Szabados operators. We prove that this sequence enjoys the variation detracting property. The convergence in variation of \({{{(R_{n}f)}_{n\geqq1}}}\) to f is also proved. A generalization in Kantorovich sense is constructed and bo...
Article
The paper deals with a class of linear positive operators expressed by q-series. By using modulus of smoothness an upper bound of approximation error is determined. We identify functions for which these operators provide uniform approximation over noncompact intervals. A particular case is delivered.
Article
Abstract Considering a general class of discrete linear positive operators, by using a one-to-one function, we associate to the class mentioned above a new sequence of operators. Our aim is to establish the transfer of approximation properties on this construction. The study is carried out in a weighted space and our results are materialized in obt...
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A new property of moduli of smoothness associated to functions belonging to some certain spaces is revealed. In terms of statistical convergence, we determine the behavior of these special functions at the point delta = 0. In this respect, Peetre's K-functional is also investigated.
Article
The topic of the present paper are certain approximation operators acting on the space of continous functions on [0,+∞) having polynomial growth. The operators which were defined by Jain in 1972 are based on a probability distribution which is called generalized Poisson distribution. As a main result we derive a complete asymptotic expansion for th...
Article
Starting from a general sequence of linear positive operators of discrete type we indicate a method to associate its an integral extension in Kantorovich sense. Numerous special cases are highlighted. Approximation properties of this extension are stated. Our goal is to show how such properties can be inherited from the discrete process to the inte...
Article
The paper deals with a general class of linear positive approximation processes designed using series. For continuous and bounded functions defined on unbounded interval we give rate of convergence in terms of the usual modulus of smoothness. The main goal is to identify functions for which these operators provide uniform approximation over unbound...
Article
In this paper we study a class of integral type positive linear operators depending on a parameter β,0⩽β<1β,0⩽β<1. Approximation properties of this class are explored: the rate of convergence in terms of the usual moduli of smoothness is given, the uniform approximation over unbounded intervals is established, the convergence in certain weighted sp...
Article
The present paper deals with the approximation of Bézier variants of Baskakov-Kantorovich operators Vn,a in the case 0 < α < 1. Pointwise approximation properties of the operators Vn,α are studied. A convergence theorem of this type approximation for locally bounded functions is established. This convergence theorem subsumes the approximation of fu...
Article
This work focuses on a class of linear positive operators of discrete type. We present the relationship between the local smoothness of functions and the local approximation. Also, the degree of approximation in terms of the moduli of smoothness is established, and the statistical convergence of the sequence is studied. Copyright © 2013 John Wiley...
Article
The main goal of the article is to introduce a class of double complex linear operators of integral type. The technique is based by extension into the complex domain of a real positive approximation process. Involving the first modulus of continuity, we investigate their geometric and approximation properties. The statistical convergence of our seq...
Article
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This paper aims to two-dimensional extension of some univariate positive approximation processes expressed by series. To be easier to use, we also modify this extension into finite sums. With respect to these two new classes designed, we investigate their approximation properties in polynomial weighted spaces. The rate of convergence is established...
Article
We introduce a class of double-complex integral linear operators. Some geometric properties are investigated and a statistical approximation theorem is obtained. In a particular case, our operators turn into the complex Picard operators.
Article
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Starting from a general sequence of linear and positive operators of summation integral type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically pointwise convergent, we prove that the p...
Article
Starting from a general sequence of linear and positive operators of discrete type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically uniform convergent, we prove that the property is i...
Article
The paper deals with a sequence of linear positive operators introduced via q-Calculus. We give a generalization in Kantorovich sense of its involving qR-integrals. Both for discrete operators and for integral operators we study the error of approximation for bounded functions and for functions having a polynomial growth. The main tools consist of...
Article
We propose a class of linear positive operators based on q-integers. These operators depend on a non-negative parameter and represent a generalization of the classical Bleimann, Butzer and Hahn operators. Approximation properties are presented and bounds of the error of approximation are established.
Chapter
This survey paper is focused on linear positive operators having the degree of exactness null and fixing the monomial of the second degree. The starting point is represented by J.P. King’s paper appearing in 2003. Our first aim is to sum up results obtained in the past five years. The second aim is to present a general class of discretizations foll...
Article
By using q-calculus, we construct Szász type operators in King’s sense, i.e., the operators preserve the first and the third test function of the Bohman-Korovkin theorem. The rate of local and global convergence is obtained in the frame of weighted spaces. The statistical approximation property of our operators is also revealed.
Article
This paper is concerned with a generalization in q-Calculus of Stancu operators. Involving modulus of continuity and Lipschitz type maximal function, we give estimates for the rate of convergence. A probabilistic approach is presented and approximation properties are established.
Article
Our goal is to present approximation theorems for sequences of positive linear operators defined on C(X), where X is a compact metric space. Instead of the uniform convergence we use the statistical convergence. Examples and special cases are also provided.
Article
In this paper we present a general class of linear positive operators of discrete type reproducing the third test function of Korovkin theorem. In a certain weighted space it forms an approximation process. A Voronovskaja-type result is established and particular cases are analyzed.
Article
This note is focused upon positive linear operators which preserve the quadratic test function. By using contraction principle, we study the convergence of their iterates.
Article
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In this note we study the limit of iterates of Lupaş q-analogue of the Bernstein operators. Also, we introduce a new class of q-Bernstein-type operators which fix certain polynomials. Both qualitative and quantitative results are established.
Article
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The concern of this note is to introduce a general class of lin-ear positive operators of discrete type acting on the space of real valued functions defined on a plane domain. These operators preserve some test functions of Bohman-Korovkin theorem. Following our technique, as a particular class, a modified variant of the bivariate Bernstein-Chlodov...
Article
This work is focused upon the study of a general class of linear positive operators of discrete type. We show that, under suitable assumptions, the sequence enjoys the variation detracting property.
Article
Full-text available
The paper centers around a pair of sequences of linear positive operators. The former has the degree of exactness one and the latter has its degree of exactness null, but, as a novelty, it reproduces the third test function of Korovkin theorem. Quantitative estimates of the rate of convergence are given in different function spaces traveling from c...
Article
The aim of the present paper is to point out basic results concerning the approximation of functions by using linear positive operators. We indicate the main research directions of this field and some of the most remarkable results obtained in the last half-century. Our presentation will bring to light classical and recent results in Korovkin-type...
Article
In the present paper we define a general class Bn,α, α ≧ 1, of Durrmeyer-Bézier type of linear positive operators. Our main aim is to estimate the rate of pointwise convergence for functions f at those points x at which the one-sided limits f(x+) and f(x-) exist. As regards these functions defined on an interval J certain conditions are required. W...
Article
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The focus of the paper is to study a class of linear positive operators constructed by using a quasi-scaling type function. Jackson type inequalities are established in the framework of dierent function spaces.
Article
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The present paper focuses on a class of linear positive operators intro-duced by G. Mastroianni. An integral extension in Kantorovich sense is defined and approximation properties of these two sequences are established in different normed spaces.
Article
The Bernstein polynomial approximation process of discrete type defined for every function f belonging to the space C([0, 1]) by \(\left( {{B_n}f} \right)\left( x \right) = \sum\nolimits_{k = 0}^n {{p_{n,k}}} \left( x \right)f\left( {k/n} \right)\), where $${p_{n,k}}\left( x \right) = \left( {\frac{n}{k}} \right){x^k}{\left( {1 - x} \right)^{n - k}...
Article
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In this paper we are dealing with a general class of positive approximation processes of discrete type expressed in series. We modify them into finite sums and investigate their ap- proximation properties in weighted spaces of continuous functions. Some special cases are also revealed.
Article
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In this paper we are concerned with a general class of positive linear operators of discrete type. Based on the results of the weakly Picard operators theory our aim is to study the convergence of the iterates of the defined operators and some approximation properties of our class as well. Some special cases in connection with binomial type operato...
Article
The present paper focuses on two approaches. Firstly, by using the contraction principle, we give a method for obtaining the limit of iterates of a class of linear positive operators. This general method is applied in studying three sequences of modified Bernstein type operators. Secondly, we define a generalization of Goodman-Sharma operators. We...
Article
In the present paper, we introduce a general class of positive operators of discrete type acting on the space of real valued functions defined on a plane domain. Based on the weakly Picard operators and the contraction principle, our aim is to study the convergence of the iterates of our defined operators. Also, some approximation properties of thi...
Article
In this paper we deal with a sequence of positive linear operatorsR [β]n approximating functions on the unbounded interval [0, t8) which were firstly used by K. Balázs and J. Szabados. We give pointwise estimates in the framework of polynomial weighted function spaces. Also we establish a Voronovskaja type theorem in the same weighted spaces for K...
Article
Based on a probabilistic theory, the paper contains local estimates of the rate of convergence for a contraction C o -semigroup. Simultaneously, a class of linear positive operators of Feller-Stancu type is introduced, and the local and global rate of convergence for continuous functions are studied.
Article
Full-text available
In this paper we construct a linear and positive approximation process of discrete type which includes as a particular case the Meyer-K¨ onig and Zeller operators. Based on several inequalities we prove that the sequence converges to the identity operator. We obtain inequalities regarding estimations of the remainder which are given by using the mo...
Article
Full-text available
We are dealing with a class of summation integral operators on an unbounded interval generated by a sequence (L n ) n≥1 of linear and positive operators. We study the degree of approximation in terms of the moduli of smoothness of first and second order. Also we present the relationship between the local smoothness of functions and the local approx...
Article
In this paper we deal with approximation by summation integral operators. We show the connections between the local smoothness of the approximated function and the rate of its local approximation. A direct theorem is obtained in a general case. Also, an inverse result is presented under certain conditions imposed on the sequence of operators, the m...
Article
We deal with a linear operator of Baskakov-type which was previously constructed by us [Rev. Anal. Numér. Théor. Approx. 26, No. 12, 3-11 (1997)] by using wavelets. Now, we estimate the order of approximation in L p -spaces (1<p≤∞) for smooth functions.
Article
We study a sequence of Bernstein-type operators, introduced and studied by D. D. Stancu in [Stud. Univ. Babeş-Bolyai, Ser. Math.-Phys. 14, 31-44 (1969; Zbl 0205.36602)]. These are depending on two parameters a and b, 0≤a≤b. We deduce first a representation by divided differences for the difference of two consecutive terms of the sequence of polynom...
Article
This paper is focused on sequences of linear positive operators, the starting point being represented by the Popoviciu-Bohman-Korovkin criterion. Our first aim is to sum up recent investigations on the statistical convergence of this type of approximation processes. The second aim is to construct a bivariate extension of Stancu discrete operators....
Article
Starting from a positive summation integral operator we present linear combinations of these operators which under definite conditions approximate a function more closely then the above operators. Also we establish a connection between the local smoothness of local Lipschitz-α(0<α≤1) functions and the local approximating property.
Article
By using probability methods we introduce a general class of Bézier type linear operators. The aim of the present paper is to estimate the rate of pointwise convergence of this class for functions of bounded variation defined on an interval J. Two cases are analyzed: Int(J)=(0,∞) and Int(J)=(0,1). In a particular case, our operators turn into the K...
Article
In this paper is introduced a general class (L k ) k∈ℤ of linear positive operators of wavelet type. The construction is based on two sequences of real numbers which verify some certain conditions. We also study some properties of the above operators. The main result consists in establishing a Jackson inequality by using the first modulus of smooth...

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