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Published: 23 February 2024

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1

Ong, Seng Huat, Choung Min Ng, Hong Keat Yap, and Hari Mohan Srivastava. "Some Probabilistic Generalizations of the Cheney–Sharma and Bernstein Approximation Operators." Axioms 11, no. 10 (October 8, 2022): 537. http://dx.doi.org/10.3390/axioms11100537.

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The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. It is also shown that the Cheney–Sharma operator is the Szász–Mirakyan operator averaged by a certain probability distribution.
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2

Ostrovska, Sofiya. "A Survey of Results on the Limit -Bernstein Operator." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/159720.

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The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit -Bernstein operator related to the approximation theory. A complete bibliography is supplied.
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3

Kajla, Arun, and Dan Miclǎuş. "Modified Bernstein–Durrmeyer Type Operators." Mathematics 10, no. 11 (May 30, 2022): 1876. http://dx.doi.org/10.3390/math10111876.

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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.
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4

Gonska, Heiner, Ioan Raşa, and Elena-Dorina Stănilă. "Lagrange-type operators associated with Uan." Publications de l'Institut Math?matique (Belgrade) 96, no. 110 (2014): 159–68. http://dx.doi.org/10.2298/pim1410159g.

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We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein-Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the derivatives of the operator images are included. Our main tool is the eigenstructure of the members of the class.
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5

Finta, Zoltán. "Approximation properties of (p, q)-Bernstein type operators." Acta Universitatis Sapientiae, Mathematica 8, no. 2 (December 1, 2016): 222–32. http://dx.doi.org/10.1515/ausm-2016-0014.

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AbstractWe introduce a new generalization of the q-Bernstein operators involving (p, q)-integers, and we establish some direct approximation results. Further, we define the limit (p, q)-Bernstein operator, and we obtain its estimation for the rate of convergence. Finally, we introduce the (p, q)-Kantorovich type operators, and we give a quantitative estimation.
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6

ACU, ANA MARIA, and P. N. AGRAWAL. "Better approximation of functions by genuine Bernstein-Durrmeyer type operators." Carpathian Journal of Mathematics 35, no. 2 (2019): 125–36. http://dx.doi.org/10.37193/cjm.2019.02.01.

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The main object of this paper is to construct a new genuine Bernstein-Durrmeyer type operators which have better features than the classical one. Some direct estimates for the modified genuine Bernstein-Durrmeyer operator by means of the first and second modulus of continuity are given. An asymptotic formula for the new operator is proved. Finally, some numerical examples with illustrative graphics have been added to validate the theoretical results and also compare the rate of convergence.
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7

Finta, Zoltan. "A generalization of the Lupaș \(q\)-analogue of the Bernstein operator." Journal of Numerical Analysis and Approximation Theory 45, no. 2 (December 9, 2016): 147–62. http://dx.doi.org/10.33993/jnaat452-1090.

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We introduce a Stancu type generalization of the Lupaș \(q\)-analogue of the Bernstein operator via the parameter \(\alpha\). The construction of our operator is based on the generalization of Gauss identity involving \(q\)-integers. We establish the convergence of our sequence of operators in the strong operator topology to the identity, estimating the rate of convergence by using the second order modulus of smoothness. For \(\alpha\) and \(q\) fixed, we study the limit operator of our sequence of operators taking into account the relationship between two consecutive terms of the constructed sequence of operators.
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8

Özger, Faruk, Ekrem Aljimi, and Merve Temizer Ersoy. "Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators." Mathematics 10, no. 12 (June 11, 2022): 2027. http://dx.doi.org/10.3390/math10122027.

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An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials in computational disciplines, we propose a new generalization of Bernstein–Kantorovich operators involving shape parameters λ, α and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program.
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9

Özalp Güller, Özge, Ecem Acar, and Sevilay Kırcı Serenbay. "Nonlinear Bivariate Bernstein–Chlodowsky Operators of Maximum Product Type." Journal of Mathematics 2022 (August 8, 2022): 1–11. http://dx.doi.org/10.1155/2022/4742433.

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The positive nonlinear operators with maximum and product were introduced by Bede. In this study, nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators is defined and the approximation properties are investigated with the help of new definitions. In this paper, it was aimed that the order of approximation obtained with the nonlinear maximum product type of operator sequences would be better than the degree of approximation of the known linear operator sequences.
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10

Usta, Fuat, Mohammad Mursaleen, and İbrahim Çakır. "Approximation properties of Bernstein-Stancu operators preserving e−2x." Filomat 37, no. 5 (2023): 1523–34. http://dx.doi.org/10.2298/fil2305523u.

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Bernstein-Stancu operators are one of the most powerful tool that can be used in approximation theory. In this manuscript, we propose a new construction of Bernstein-Stancu operators which preserve the constant and e?2x, x > 0. In this direction, the approximation properties of this newly defined operators have been examined in the sense of different function spaces. In addition to these, we present the Voronovskaya type theorem for this operators. At the end, we provide two computational examples to demonstrate that the new operator is an approximation procedure.
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11

Badea, Catalin. "Bernstein Polynomials and Operator Theory." Results in Mathematics 53, no. 3-4 (June 29, 2009): 229–36. http://dx.doi.org/10.1007/s00025-008-0333-1.

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12

Srivastava, Hari, Faruk Özger, and S. Mohiuddine. "Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ." Symmetry 11, no. 3 (March 2, 2019): 316. http://dx.doi.org/10.3390/sym11030316.

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We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter λ ∈ [ - 1 , 1 ] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ -Bernstein operators and study their approximation behaviors.
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13

Ozdemir, M. Kemal, Ayhan Esi, and Ayten Esi. "Bernstein operator of rough I-core of triple sequences." ITM Web of Conferences 22 (2018): 01060. http://dx.doi.org/10.1051/itmconf/20182201060.

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We introduce and study some basic properties of Bernstein-Stancu polynomials of rough I-convergent of triple sequence spaces and also study the set of all Bernstein-Stancu polynomials of rough I-limits of a triple sequence spaces and relation between analytic ness and Bernstein-Stancu polynomials of rough I-core of a triple sequence spaces.
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14

Miclăuş, Dan. "The generalization of the Bernstein operator on any finite interval." Georgian Mathematical Journal 24, no. 3 (September 1, 2017): 447–53. http://dx.doi.org/10.1515/gmj-2016-0043.

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AbstractThe paper presents the generalized form of the Bernstein operator associated with any real-valued function {f\colon[a,b]\to\mathbb{R}}. For this generalized Bernstein operator, we study the qualitative and quantitative aspects concerning uniform convergence, order of approximation and asymptotic behavior.
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15

Mishra, Ravendra Kumar, Sudesh Kumar Garg, Rupa Rani Sharma, and Priyanka Sharma. "Estimation Features by Transformed Bernstein kind Polynomials." International Journal of Experimental Research and Review 32 (August 30, 2023): 110–14. http://dx.doi.org/10.52756/ijerr.2023.v32.008.

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In our extensive study of literature, we delved into the multifarious manifestations of discrete operator transformations. These transformations are pivotal in mathematical analysis, especially concerning Lebesgue integral equations. Our investigation led us to corroborate the findings of Acu, Heilmann and Lorentz particularly in the context of functions normed under the L1-norm. Generalization was a key facet of our research, wherein we probed deeper into these operators' behaviors. This endeavor yielded a profound result: the derivation of a global asymptotic formula, providing invaluable insight into the long-term trends exhibited by these operators. Such formulae are instrumental in predicting the operators' behaviors over an extended span. Furthermore, our exploration unveiled a plethora of findings related to these generalized operators. We meticulously computed various moments, shedding light on the statistical characteristics of these transformations. This included an investigation into convergence properties, essential for understanding the stability and reliability of the operators in question. One of the most noteworthy contributions of our study is the elucidation of pointwise approximation and direct results. These findings offer practical applications, allowing for precise and efficient approximations in practical scenarios. This is particularly significant in fields where these operators are routinely employed, such as signal processing, numerical analysis, and scientific computing. In essence, our research has not only confirmed the foundational work of Acu, Heilmann and Lorentz but has also expanded the horizons of knowledge surrounding discrete operator transformations, offering a wealth of insights and practical implications for a wide range of mathematical and computational applications.
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16

Goodman, T. N. T., and A. Sharma. "A Bernstein-Schoenberg Type Operator: Shape Preserving and Limiting Behaviour." Canadian Journal of Mathematics 47, no. 5 (October 1, 1995): 959–73. http://dx.doi.org/10.4153/cjm-1995-050-8.

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AbstractUsing a new B-spline basis due to Dahmen, Micchelli and Seidel, we construct a univariate spline approximation operator of Bernstein-Schoenberg type. We show that it shares all the shape preserving properties of the usual Bernstein-Schoenberg operator and we derive a Voronovskaya type asymptotic error estimate.
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17

Aslan, Reşat, and Aydın İzgi. "Approximation by One and Two Variables of the Bernstein-Schurer-Type Operators and Associated GBS Operators on Symmetrical Mobile Interval." Journal of Function Spaces 2021 (May 3, 2021): 1–12. http://dx.doi.org/10.1155/2021/9979286.

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In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’s K -functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.
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18

Aral, Ali, and Hasan Erbay. "Comparison of Two-Parameter Bernstein Operator and Bernstein–Durrmeyer Variants." Bulletin of the Iranian Mathematical Society 44, no. 6 (July 14, 2018): 1471–84. http://dx.doi.org/10.1007/s41980-018-0101-2.

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19

Garg, Tarul, Nurhayat İspir, and P. N. Agrawal. "Bivariate q-Bernstein–Chlodowsky–Durrmeyer type operators and the associated GBS operators." Asian-European Journal of Mathematics 13, no. 05 (April 4, 2019): 2050091. http://dx.doi.org/10.1142/s1793557120500916.

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This paper deals with the approximation properties of the [Formula: see text]-bivariate Bernstein–Chlodowsky operators of Durrmeyer type. We investigate the approximation degree of the [Formula: see text]-bivariate operators for continuous functions in Lipschitz space and also with the help of partial modulus of continuity. Further, the Generalized Boolean Sum (GBS) operator of these bivariate [Formula: see text]–Bernstein–Chlodowsky–Durrmeyer operators is introduced and the rate of convergence in the Bögel space of continuous functions by means of the Lipschitz class and the mixed modulus of smoothness is examined. Furthermore, the convergence and its comparisons are shown by illustrative graphics for the [Formula: see text]-bivariate operators and the associated GBS operators to certain functions using Maple algorithms.
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20

Finta, Zoltán. "Approximation by limit q-Bernstein operator." Acta Universitatis Sapientiae, Mathematica 5, no. 1 (February 1, 2013): 39–46. http://dx.doi.org/10.2478/ausm-2014-0003.

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21

Tachev, Gancho. "On multiplicativity of the Bernstein operator." Computers & Mathematics with Applications 62, no. 8 (October 2011): 3236–40. http://dx.doi.org/10.1016/j.camwa.2011.08.038.

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22

Szabados, J. "On a quasi-interpolating Bernstein operator." Journal of Approximation Theory 196 (August 2015): 1–12. http://dx.doi.org/10.1016/j.jat.2015.02.009.

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23

Cooper, Shaun, and Shayne Waldron. "The Eigenstructure of the Bernstein Operator." Journal of Approximation Theory 105, no. 1 (July 2000): 133–65. http://dx.doi.org/10.1006/jath.2000.3464.

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24

BARBOSU, DAN. "Some identities involving divided differences." Creative Mathematics and Informatics 24, no. 2 (2015): 107–11. http://dx.doi.org/10.37193/cmi.2015.02.14.

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To study approximation properties of linear positive operators various identities involving divided differences are used. The aim of this note is to present two types of such kind of identities. The first one was used by Abel and Ivan [Abel, U. and Ivan, M., Some identities for the operator of Bleimamm, Butzer and Hahn involving divided differences, Calcolo, 36 (1999), 143–160; Abel, U. and Ivan, M., New representation of the remainder in the Bernstein approximation, J. Math. Anal. Appl., 381 (2011), No. 2, 952–956] to derive approximation properties of Bleimann, Butzer and Hahn (BBH) operators from the corresponding properties of the classical Bernstein operators. The second type of identifies can be used to derive some approximation properties of the BBH operators from the properties of some Stancu type operators.
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25

de la Cal, Jesús, and Ana M. Valle. "Best constants for tensor products of Bernstein, Szász and Baskakov operators." Bulletin of the Australian Mathematical Society 62, no. 2 (October 2000): 211–20. http://dx.doi.org/10.1017/s0004972700018682.

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We consider tensor product operators and discuss their best constants in preservation inequalities concerning the usual moduli of continuity. In a previous paper, we obtained lower and upper bounds on such constants, under fairly general assumptions on the operators. Here, we concentrate on the l∞-modulus of continuity and three celebrated families of operators. For the tensor product of k identical copies of the Bernstein operator Bn, we show that the best uniform constant coincides with the dimension k when k ≥ 3, while, in case k = 2, it lies in the interval [2, 5/2] but depends upon n. Similar results also hold when Bn is replaced by a univariate Szász or Baskakov operator. The three proofs follow the same pattern, a crucial ingredient being some special properties of the probability distributions involved in the mentioned operators, namely: the binomial, Poisson, and negative binomial distributions.
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26

RASA, IOAN. "Estimates for the semigroup associated with Bernstein-Schnabl operators." Carpathian Journal of Mathematics 28, no. 1 (2012): 157–62. http://dx.doi.org/10.37193/cjm.2012.01.02.

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We consider a differential operator of the form Au(x) = α(x)u”(x), 0 ≤ x ≤ 1. Under suitable assumptions, it generates a semigroup (T(t))t≥0 which can be approximated by iterates of the Bernstein-Schnabl operators. We obtain quantitative results concerning the behaviour of T(t) when t → 0 + and t → +∞.
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27

Auad, Alaa Adnan, Mohammed A. Hilal, and Nihad Shareef Khalaf. "Best Approximation of Unbounded Functions by Modulus of Smoothness." European Journal of Pure and Applied Mathematics 16, no. 2 (April 30, 2023): 944–52. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4730.

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In this paper, we study the approximation of unbounded functions in a weighted space by modulus of smoothness using various linear operators. We establish direct theorems for such approximations and analyze the properties of the modulus of smoothness within the same space. Specifically, we investigate the behavior of the modulus of smoothness under different types of linear operators, including the Bernstein-Durrmeyer operator, the Fejer operator, and the Jackson operator. We also provide a detailed analysis of the convergence rate of these operators. Furthermore, we discuss the relationship between the modulus of smoothness and the Lipschitz constant of a function. Our findings have important implications for the field of approximation theory and may help to inform future research in this area.
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28

Liu, Yu-Jie, Wen-Tao Cheng, Wen-Hui Zhang, and Pei-Xin Ye. "Approximation Properties of the Blending-Type Bernstein–Durrmeyer Operators." Axioms 12, no. 1 (December 21, 2022): 5. http://dx.doi.org/10.3390/axioms12010005.

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We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.
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29

Shehab, Mustafa K. "Approximation using a modified type of Bernstein operators." BASRA JOURNAL OF SCIENCE 40, no. 3 (December 1, 2022): 558–69. http://dx.doi.org/10.29072/basjs.20220303.

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A new generalization for the Bernstein-Kantorovich operator with a parameter is proposed in this study. First, we prove the Korovkin type approximation theorem, then we provide the Voronovskaja type theorem for our generalization, demonstrating that the order of approximation is improved, making the approximation by our operators better than the original Kantorovich, and finally, we provide some numerical data for two test functions to support the study.
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30

Mirotin, A. R. "Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus." Filomat 34, no. 4 (2020): 1105–15. http://dx.doi.org/10.2298/fil2004105m.

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We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove the formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.
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31

Mirotin, A. R. "Perturbation determinants on Banach spaces and operator differentiability for Hirsch functional calculus." Filomat 34, no. 4 (2020): 1105–15. http://dx.doi.org/10.2298/fil2004105m.

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We consider a perturbation determinant for pairs of nonpositive (in a sense of Komatsu) operators on Banach space with nuclear difference and prove the formula for the logarithmic derivative of this determinant. To this end the Frechet differentiability of operator monotonic (negative complete Bernstein) functions of negative and nonpositive operators on Banach spaces is investigated. The results may be regarded as a contribution to the Hirsch functional calculus.
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32

Aleksandrov, A. B., and V. V. Peller. "Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities." Indiana University Mathematics Journal 59, no. 4 (2010): 1451–90. http://dx.doi.org/10.1512/iumj.2010.59.4345.

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33

Adell, Jose A., and Ana Perez-Palomares. "Stochastic orders in preservation properties by Bernstein-type operators." Advances in Applied Probability 31, no. 2 (June 1999): 492–509. http://dx.doi.org/10.1239/aap/1029955144.

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In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.
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34

Adell, Jose A., and Ana Perez-Palomares. "Stochastic orders in preservation properties by Bernstein-type operators." Advances in Applied Probability 31, no. 02 (June 1999): 492–509. http://dx.doi.org/10.1017/s0001867800009204.

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In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.
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35

Rai, Deepmala, and N. Subramanian. "Sliding Window Rough measurable function on $I-$ core of triple sequences of Bernstein operator." Journal of the Indonesian Mathematical Society 25, no. 3 (October 31, 2019): 183–93. http://dx.doi.org/10.22342/jims.25.3.687.183-193.

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We introduce sliding window rough $I-$ core and study some basic properties of Bernstein polynomials of rough $I-$ convergent of triple sequence spaces and also study the set of all Bernstein polynomials of sliding window of rough $I-$ limits of a triple sequence spaces and relation between analytic ness and Bernstein polynomials of sliding window of rough $I-$ core of a triple sequence spaces.
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36

Acu, Ana, and Ioan Raşa. "On the composition and decomposition of positive linear operators (VII)." Applicable Analysis and Discrete Mathematics 15, no. 1 (2021): 213–32. http://dx.doi.org/10.2298/aadm191103006a.

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In the present paper we study the compositions of the piecewise linear interpolation operator S?n and the Beta-type operator B?n, namely An:= S?n ?B?n and Gn := B?n ? S?n. Voronovskaya type theorems for the operators An and Gn are proved, substantially improving some corresponding known results. The rate of convergence for the iterates of the operators Gn and An is considered. Some estimates of the differences between An, Gn, Bn and S?n, respectively, are given. Also, we study the behaviour of the operators An on the subspace of C[0,1] consisting of all polygonal functions with nodes {0, 1/2,..., n-1/n,1}. Finally, we propose to the readers a conjecture concerning the eigenvalues of the operators An and Gn. If true, this conjecture would emphasize a new and strong relationship between Gn and the classical Bernstein operator Bn.
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37

Bede, Barnabás, Lucian Coroianu, and Sorin G. Gal. "Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind." International Journal of Mathematics and Mathematical Sciences 2009 (2009): 1–26. http://dx.doi.org/10.1155/2009/590589.

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Starting from the study of theShepard nonlinear operator of max-prod typeby Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324–326, theBernstein max-prod-type operatoris introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form (with an unexplicit absolute constant ) and the question of improving the order of approximation is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions including, for example, that of concave functions, we find the order of approximation , which for many functions is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.
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38

ÇETIN, NURSEL. "A new complex generalized Bernstein-Schurer operator." Carpathian Journal of Mathematics 37, no. 1 (February 5, 2021): 81–89. http://dx.doi.org/10.37193/cjm.2021.01.08.

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"In this paper, we consider the complex form of a new generalization of Bernstein-Schurer operators. We obtain some quantitative upper estimates for the approximation of these operators attached to analytic functions. Moreover, we prove that these operators preserve some properties of the original function such as univalence, starlikeness, convexity and spirallikeness."
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39

Esi, Ayhan, and Serkan Araci. "Lacunary statistical convergence of Bernstein operator sequences." International Journal of ADVANCED AND APPLIED SCIENCES 4, no. 11 (November 2017): 78–80. http://dx.doi.org/10.21833/ijaas.2017.011.011.

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40

BIROU, MARIUS. "Blending surfaces generated using the Bernstein operator." Creative Mathematics and Informatics 21, no. 1 (2012): 35–40. http://dx.doi.org/10.37193/cmi.2012.01.05.

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In this paper we construct blending surfaces using the univariate Bernstein operator. The surfaces have the properties that they stay on a curve (the border of the surfaces domain) and have a fixed height in a point from the domain. The surfaces are generated using a curve network, instead of the control points from the case of classical Bezier surfaces. We study the monotonicity and we give conditions to obtain concave surfaces.
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41

Cooper, Shaun, and Shayne Waldron. "The Diagonalisation of the Multivariate Bernstein Operator." Journal of Approximation Theory 117, no. 1 (July 2002): 103–31. http://dx.doi.org/10.1006/jath.2002.3683.

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42

Mir, A., and A. Hussain. "Operator Preserving Bernstein-Type Inequalities between Polynomials." Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 58, no. 5 (October 2023): 347–56. http://dx.doi.org/10.3103/s1068362323050059.

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43

Özkan, Esma Yıldız, and Gözde Aksoy. "Approximation by Tensor-Product Kind Bivariate Operator of a New Generalization of Bernstein-Type Rational Functions and Its GBS Operator." Mathematics 10, no. 9 (April 22, 2022): 1418. http://dx.doi.org/10.3390/math10091418.

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We introduce a tensor-product kind bivariate operator of a new generalization of Bernstein-type rational functions and its GBS (generalized Boolean sum) operator, and we investigate their approximation properties by obtaining their rates of convergence. Moreover, we present some graphical comparisons visualizing the convergence of tensor-product kind bivariate operator and its GBS operator.
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44

Agrawal, Purshottam, Dharmendra Kumar, and Behar Baxhaku. "On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers." Journal of Numerical Analysis and Approximation Theory 51, no. 1 (September 17, 2022): 3–36. http://dx.doi.org/10.33993/jnaat511-1244.

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In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.
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45

COROIANU, LUCIAN, and SORIN G. GAL. "CLASSES OF FUNCTIONS WITH IMPROVED ESTIMATES IN APPROXIMATION BY THE MAX-PRODUCT BERNSTEIN OPERATOR." Analysis and Applications 09, no. 03 (July 2011): 249–74. http://dx.doi.org/10.1142/s0219530511001856.

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In this paper, we find large classes of positive functions, others than those in [1], having even a Jackson-type estimate, ω1(f;1/n), in approximation by the nonlinear max-product Bernstein operator. The uniform estimate of the order O[nω1(f;1/n)2 + ω1(f;1/n)] is achieved, while near to the endpoints 0 and 1, the better pointwise estimate of the order [Formula: see text] is obtained. Finally, we prove that besides the preservation of quasi-convexity found in [1], the nonlinear max-product Bernstein operator preserves the quasi-concavity too.
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46

Velmurugan, S., and N. Subramanian. "Bernstein Operator of Rough λ-statistically and ρ Cauchy Sequences Convergence on Triple Sequence Spaces." Journal of the Indian Mathematical Society 85, no. 1-2 (January 4, 2018): 256. http://dx.doi.org/10.18311/jims/2018/15896.

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<p>In this article, using the concept of natural density, we introduce the notion of Bernstein polynomials of rough λ−statistically and ρ−Cauchy triple sequence spaces. We define the set of Bernstein polynomials of rough statistical limit points of a triple sequence spaces and obtain to λ−statistical convergence criteria associated with this set. We examine the relation between the set of Bernstein polynomials of rough λ−statistically and ρ− Cauchy triple sequences.</p><p> </p><p> </p>
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47

Cătinaş, Teodora. "A Review on Some Linear Positive Operators Defined on Triangles." Symmetry 14, no. 9 (September 8, 2022): 1880. http://dx.doi.org/10.3390/sym14091880.

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We consider results regarding Bernstein and Cheney–Sharma-type operators that interpolate functions defined on triangles with straight and curved sides and we introduce a new Cheney–Sharma-type operator for the triangle with one curved side, highlighting the symmetry between the methods. We present some properties of the operators, their products and Boolean sums and some results regarding the remainders of the corresponding approximation formulas, using modulus of continuity and Peano’s theorem. Additionally, we consider some numerical examples to show the approximation properties of the given operators.
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48

Ostrovska, Sofiya. "The Functional-Analytic Properties of the Limitq-Bernstein Operator." Journal of Function Spaces and Applications 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/280314.

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The limitq-Bernstein operatorBq,0<q<1, emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in theq-boson theory to describe the energy distribution in aq-analogue of the coherent state. Lately, the limitq-Bernstein operator has been widely under scrutiny, and it has been shown thatBqis a positive shape-preserving linear operator onC[0,1]with∥Bq∥=1. Its approximation properties, probabilistic interpretation, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties ofBqare studied. Our main result states that there exists an infinite-dimensional subspaceMofC[0,1]such that the restrictionBq|Mis an isomorphic embedding. Also we show that each such subspaceMcontains an isomorphic copy of the Banach spacec0.
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Karaisa, Ali, and Uğur Kadak. "On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem." Filomat 31, no. 12 (2017): 3749–60. http://dx.doi.org/10.2298/fil1712749k.

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Upon prior investigation on statistical convergence of fuzzy sequences, we study the notion of pointwise ??-statistical convergence of fuzzy mappings of order ?. Also, we establish the concept of strongly ??-summable sequences of fuzzy mappings and investigate some inclusion relations. Further, we get an analogue of Korovkin-type approximation theorem for fuzzy positive linear operators with respect to ??-statistical convergence. Lastly, we apply fuzzy Bernstein operator to construct an example in support of our result.
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50

Demkiv, I. I. "On properties of Bernstein-type operator polynomials that approximate the Urysohn operator." Ukrainian Mathematical Journal 56, no. 9 (September 2004): 1391–402. http://dx.doi.org/10.1007/s11253-005-0123-9.

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