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Автор: Grafiati

Опубліковано: 26 жовтня 2023

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1

Lyamina, O. S. "ON NORMS AND CERTAIN CHARACTERISTICS OF TRIGONOMETRIC APPROXIMATION BY BASKAKOV OPERATORS." Vestnik of Samara University. Natural Science Series 18, no. 9 (June 9, 2017): 41–51. http://dx.doi.org/10.18287/2541-7525-2012-18-9-41-51.

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Анотація:

The article covers actual question in the theory of approximations. It researches approximative opportunities of concrete approximating structures. In the article one of the actively studied in recent times types of approximating operators — Baskakov's trigonometric operators. Some characteristics of these operators are being investigated: norms and approximation constants, assessment and improved. In particular, the assessment of their difference is obtained.

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2

Duma, Adrian, and Cristian Vladimirescu. "Approximation structures and applications to evolution equations." Abstract and Applied Analysis 2003, no. 12 (2003): 685–96. http://dx.doi.org/10.1155/s1085337503301010.

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Анотація:

We discuss various properties of the nonlinearA-proper operators as well as a generalized Leray-Schauder principle. Also, a method of approximating arbitrary continuous operators byA-proper mappings is described. We construct, via appropriate Browder-Petryshyn approximation schemes, approximative solutions for linear evolution equations in Banach spaces.

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3

Tang, Weidong, Jinzhao Wu, and Dingwei Zheng. "On Fuzzy Rough Sets and Their Topological Structures." Mathematical Problems in Engineering 2014 (2014): 1–17. http://dx.doi.org/10.1155/2014/546372.

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Анотація:

The core concepts of rough set theory are information systems and approximation operators of approximation spaces. Approximation operators draw close links between rough set theory and topology. This paper is devoted to the discussion of fuzzy rough sets and their topological structures. Fuzzy rough approximations are further investigated. Fuzzy relations are researched by means of topology or lower and upper sets. Topological structures of fuzzy approximation spaces are given by means of pseudoconstant fuzzy relations. Fuzzy topology satisfying (CC) axiom is investigated. The fact that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies satisfying (CC) axiom is proved, the concept of fuzzy approximating spaces is introduced, and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained, which illustrates that we can research fuzzy relations or fuzzy approximation spaces by means of topology and vice versa. Moreover, fuzzy pseudoclosure operators are examined.

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4

Anastassiou, George A. "Multivariate and abstract approximation theory for Banach space valued functions." Demonstratio Mathematica 50, no. 1 (August 28, 2017): 208–22. http://dx.doi.org/10.1515/dema-2017-0020.

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Анотація:

Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

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5

Qasim, Mohd, M. Mursaleen, Asif Khan, and Zaheer Abbas. "Approximation by Generalized Lupaş Operators Based on q-Integers." Mathematics 8, no. 1 (January 2, 2020): 68. http://dx.doi.org/10.3390/math8010068.

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Анотація:

The purpose of this paper is to introduce q-analogues of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing, and unbounded function ρ . Depending on the selection of q, these operators provide more flexibility in approximation and the convergence is at least as fast as the generalized Lupaş operators, while retaining their approximation properties. For these operators, we give weighted approximations, Voronovskaja-type theorems, and quantitative estimates for the local approximation.

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6

Yuan Wu, Pei. "Approximation by partial isometries." Proceedings of the Edinburgh Mathematical Society 29, no. 2 (June 1986): 255–61. http://dx.doi.org/10.1017/s0013091500017624.

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Анотація:

Let B(H) be the algebra of bounded linear operators on a complex separable Hilbert space H. The problem of operator approximation is to determine how closely each operator T ∈B(H) can be approximated in the norm by operators in a subset L of B(H). This problem is initiated by P. R. Halmo [3] when heconsidered approximating operators by the positive ones. Since then, this problem has been attacked with various classes L: the class of normal operators whose spectrum is included in a fixed nonempty closed subset of the complex plane [4], the classes of unitary operators [6] and invertible operators [1]. The purpose of this paper is to study the approximation by partial isometries.

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7

Zhao, Tao, and Zhenbo Wei. "On Characterization of Rough Type-2 Fuzzy Sets." Mathematical Problems in Engineering 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/4819353.

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Анотація:

Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the generalized Pawlak approximation space are also introduced. Some basic properties of the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators are discussed. The connections between special crisp binary relations and generalized rough type-2 fuzzy approximation operators are further examined. The axiomatic characterization of generalized rough type-2 fuzzy approximation operators is also presented. Finally, the attribute reduction of type-2 fuzzy information systems is investigated.

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8

Zayed, Mohra, Shahid Ahmad Wani, and Mohammad Younus Bhat. "Unveiling the Potential of Sheffer Polynomials: Exploring Approximation Features with Jakimovski–Leviatan Operators." Mathematics 11, no. 16 (August 21, 2023): 3604. http://dx.doi.org/10.3390/math11163604.

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Анотація:

In this article, we explore the construction of Jakimovski–Leviatan operators for Durrmeyer-type approximation using Sheffer polynomials. Constructing positive linear operators for Sheffer polynomials enables us to analyze their approximation properties, including weighted approximations and convergence rates. The application of approximation theory has earned significant attention from scholars globally, particularly in the fields of engineering and mathematics. The investigation of these approximation properties and their characteristics holds considerable importance in these disciplines.

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9

KHAN, TAQSEER, MOHD SAIF, and SHUZAAT ALI KHAN. "APPROXIMATION BY GENERALIZED q-SZASZ-MIRAKJAN ´ OPERATORS." Journal of Mathematical Analysis 12, no. 6 (December 31, 2021): 9–21. http://dx.doi.org/10.54379/jma-2021-6-2.

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Анотація:

In this article, we introduce generalized q−Sz´asz-Mirakjan operators and study their approximation properties. Based on the Voronovskaja’s theorem, we obtain quantitative estimates for these operators.

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10

Crespo, José, and Francisco Javier Montáns. "Fractional Mathematical Operators and Their Computational Approximation." Mathematical Problems in Engineering 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/4356371.

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Анотація:

Usual applied mathematics employs three fundamental arithmetical operators: addition, multiplication, and exponentiation. However, for example, transcendental numbers are said not to be attainable via algebraic combination with these fundamental operators. At the same time, simulation and modelling frequently have to rely on expensive numerical approximations of the exact solution. The main purpose of this article is to analyze new fractional arithmetical operators, explore some of their properties, and devise ways of computing them. These new operators may bring new possibilities, for example, in approximation theory and in obtaining closed forms of those approximations and solutions. We show some simple demonstrative examples.

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11

Yadav, Rishikesh, Ramakanta Meher, and Vishnu Narayan Mishra. "Further Approximations of Durrmeyer Modification of Szasz-Mirakjan Operators." European Journal of Pure and Applied Mathematics 13, no. 5 (December 27, 2020): 1306–24. http://dx.doi.org/10.29020/nybg.ejpam.v13i5.3728.

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Анотація:

The main purpose of this paper is to determine the approximations of Durrmeyermodification of Szasz-Mirakjan operators, defined by Mishra et al. (Boll. Unione Mat. Ital. (2016) 8(4):297-305). We estimate the order of approximation of the operators for the functions belonging to the different spaces. Here, the rate of convergence of the said operators is established by means of the function with derivative of the bounded variation. At last, the graphical analysis is discussed to support the approximation results of the operators.

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12

Mahmudov, N. I. "Approximation by the -Szász-Mirakjan Operators." Abstract and Applied Analysis 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/754217.

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Анотація:

This paper deals with approximating properties of theq-generalization of the Szász-Mirakjan operators in the case . Quantitative estimates of the convergence in the polynomial-weighted spaces and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by theq-Szász-Mirakjan operators ( ) is of order versus 1/nfor the classical Szász-Mirakjan operators.

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13

Rao, Nadeem, та Pradeep Malik. "Α-Baskakov-Durrmeyer type operators and their approximation properties". Filomat 37, № 3 (2023): 935–48. http://dx.doi.org/10.2298/fil2303935r.

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Анотація:

In the present research article, we construct a new family of summation-integral type hybrid operators in terms of shape parameter ? ? [0, 1]. Further, basic estimates, rate of convergence and the order of approximation with the aid of Korovkin theorem and modulus of smoothness are investigated. Moreover, numerical simulation and graphical approximations are studied. For these sequences of positive linear operators, we study the local approximation results using Peetre?s K-functional, Lipschitz class and modulus of smoothness of second order. Next, we obtain the approximation results in weighted space. Lastly, A-statistical-approximation results are presented.

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14

Karahan, Döne, and Aydın İzgi. "On Approximation Properties of (p,q)-Bernstein Operators." European Journal of Pure and Applied Mathematics 11, no. 2 (April 27, 2018): 457–67. http://dx.doi.org/10.29020/nybg.ejpam.v11i2.3213.

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Анотація:

In this study, a (p,q)-analogue of Bernstein operators is introducedand approximation properties of (p,q)-Bernstein operators areinvestigated. Some basic theorems are proved. The rate of approximationby modulus of continuity is estimated.

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15

Boffi, Daniele. "Finite element approximation of eigenvalue problems." Acta Numerica 19 (May 2010): 1–120. http://dx.doi.org/10.1017/s0962492910000012.

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Анотація:

We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.

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16

Dmytryshyn, M. I. "Approximation of positive operators by analytic vectors." Carpathian Mathematical Publications 12, no. 2 (December 27, 2020): 412–18. http://dx.doi.org/10.15330/cmp.12.2.412-418.

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Анотація:

We give the estimates of approximation errors while approximating of a positive operator $A$ in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ associated with the complex degrees of positive operator. The approximation spaces are determined by $E$-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by $K$-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.

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17

Li, Zhaowen, Bin Qin, and Zhangyong Cai. "Soft Rough Approximation Operators and Related Results." Journal of Applied Mathematics 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/241485.

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Анотація:

Soft set theory is a newly emerging tool to deal with uncertain problems. Based on soft sets, soft rough approximation operators are introduced, and soft rough sets are defined by using soft rough approximation operators. Soft rough sets, which could provide a better approximation than rough sets do, can be seen as a generalized rough set model. This paper is devoted to investigating soft rough approximation operations and relationships among soft sets, soft rough sets, and topologies. We consider four pairs of soft rough approximation operators and give their properties. Four sorts of soft rough sets are investigated, and their related properties are given. We show that Pawlak's rough set model can be viewed as a special case of soft rough sets, obtain the structure of soft rough sets, give the structure of topologies induced by a soft set, and reveal that every topological space on the initial universe is a soft approximating space.

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18

Dmytryshyn, M. I. "Interpolated scales of approximation spaces for regular elliptic operators on compact manifolds." Carpathian Mathematical Publications 6, no. 1 (July 19, 2014): 26–31. http://dx.doi.org/10.15330/cmp.6.1.26-31.

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Анотація:

We define the interpolated scales of approximation spaces, generated by regular elliptic operators on compact manifolds. The appropriate Bernstein-Jackson inequalities and application to spectral approximations of regular elliptic operators are considered.

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19

Mao, Hua. "Approximation operators for semiconcepts." Journal of Intelligent & Fuzzy Systems 36, no. 4 (April 11, 2019): 3333–43. http://dx.doi.org/10.3233/jifs-18104.

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20

Bhatia, Rajendra, and Fuad Kittaneh. "Approximation by positive operators." Linear Algebra and its Applications 161 (January 1992): 1–9. http://dx.doi.org/10.1016/0024-3795(92)90001-q.

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21

Zhang, Yan-Lan, and Chang-Qing Li. "Topological structures of a type of granule based covering rough sets." Filomat 32, no. 9 (2018): 3129–41. http://dx.doi.org/10.2298/fil1809129z.

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Анотація:

Rough set theory is one of important models of granular computing. Lower and upper approximation operators are two important basic concepts in rough set theory. The classical Pawlak approximation operators are based on partition and have been extended to covering approximation operators. Covering is one of the fundamental concepts in the topological theory, then topological methods are useful for studying the properties of covering approximation operators. This paper presents topological properties of a type of granular based covering approximation operators, which contains seven pairs of approximation operators. Then, topologies are induced naturally by the seven pairs of covering approximation operators, and the topologies are just the families of all definable subsets about the covering approximation operators. Binary relations are defined from the covering to present topological properties of the topological spaces, which are proved to be equivalence relations. Moreover, connectedness, countability, separation property and Lindel?f property of the topological spaces are discussed. The results are not only beneficial to obtain more properties of the pairs of covering approximation operators, but also have theoretical and actual significance to general topology.

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22

Medegey, M. B. "ON ONE APPROXIMATION ESTIMATE." Vestnik of Samara University. Natural Science Series 17, no. 5 (June 14, 2017): 53–59. http://dx.doi.org/10.18287/2541-7525-2011-17-5-53-59.

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Анотація:

Linear operators satisfying some conditions are considered. The given operators are a particular kind of class operators (by P.P. Korovkin). For the estimate derivation the interpolation method is used, described in the works by Yu.G. Abakumov and O.N. Shestakova

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23

Abel, Ulrich, and Octavian Agratini. "On the Durrmeyer-Type Variant and Generalizations of Lototsky–Bernstein Operators." Symmetry 13, no. 10 (October 1, 2021): 1841. http://dx.doi.org/10.3390/sym13101841.

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Анотація:

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 indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube [0,1]q, q>1. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.

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24

Kazlouskaya, N. Yu, and Ya A. Rovba. "On the approximation of the | sin |s x function by rational trigonometric operators of the Fejér type." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 59, no. 2 (July 6, 2023): 95–109. http://dx.doi.org/10.29235/1561-2430-2023-59-2-95-109.

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Анотація:

Approximation by trigonometric Fourier series is a well-developed branch of the theory of approximation by polynomials. Methods of approximation by rational trigonometric Fourier series have not been researched so deeply yet. In particular, rational trigonometric operators of the Fejér type have not been used in the rational approximation with free poles. In this paper, we consider the approximation of the function | sin | , (0;2), ∈ s x s by rational trigonometric operators of the Fejér type. An integral representation of the remainder for the above-mentioned approximation is obtained. An estimate of approximations is found in the points of analyticity of the function | sin |s x under the condition that the corresponding system of rational functions is complete. It is shown that the order of uniform approximation in the case of approximation by rational Fejér functions with two geometrically different poles is higher than the order of approximation by trigonometric polynomials. As a result, an asymptotic estimation of the uniform approximation by trigonometric Fejér sums in the polynomial case is obtained.

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25

Ma, Liwen. "Some twin approximation operators on covering approximation spaces." International Journal of Approximate Reasoning 56 (January 2015): 59–70. http://dx.doi.org/10.1016/j.ijar.2014.08.003.

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26

Yaremchuk, Max, and Matthew Carrier. "On the Renormalization of the Covariance Operators." Monthly Weather Review 140, no. 2 (February 2012): 637–49. http://dx.doi.org/10.1175/mwr-d-11-00139.1.

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Анотація:

Many background error correlation (BEC) models in data assimilation are formulated in terms of a smoothing operator [Formula: see text], which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, [Formula: see text] has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of [Formula: see text]) is a computationally expensive procedure, which needs an efficient numerical approximation. In this study approximate renormalization techniques based on the Monte Carlo (MC) and Hadamard matrix (HM) methods and on the analytic approximations derived under the assumption of the local homogeneity (LHA) of [Formula: see text] are compared using realistic BEC models designed for oceanographic applications. It is shown that although the accuracy of the MC and HM methods can be improved by additional smoothing, their computational cost remains significantly higher than the LHA method, which is shown to be effective even in the zeroth-order approximation. The next approximation improves the accuracy 1.5–2 times at a moderate increase of CPU time. A heuristic relationship for the smoothing scale in two and three dimensions is proposed for the first-order LHA approximation.

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27

Yang, Kai, and Jianfeng Zhang. "Comparison between Born and Kirchhoff operators for least-squares reverse time migration and the constraint of the propagation of the background wavefield." GEOPHYSICS 84, no. 5 (September 1, 2019): R725—R739. http://dx.doi.org/10.1190/geo2018-0438.1.

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Анотація:

The Born approximation and the Kirchhoff approximation are two frameworks that are extensively used in solving seismic migration/inversion problems. Both approximations assume a linear relationship between the primary reflected/scattered data to the corresponding physical model. However, different approximations result in different behaviors. For least-squares reverse time migration (LSRTM), most of the algorithms are constructed based on Born approximation. We have constructed a pair of Kirchhoff modeling and migration operators based on the Born modeling operator and the connection between the perturbation model and the reflectivity model, and then we compared the different performances between Born and Kirchhoff operators for LSRTM. Numerical examples on Marmousi model and SEAM 2D salt model indicate that LSRTM with Kirchhoff operators is a better alternative to that with Born operators for imaging complex structures. To reduce the computational cost, we also investigate a strategy by restricting the propagation of the background wavefield to a stopping time rather than the maximum recording time. And this stopping time can be chosen as half of the maximum recording time. This computational strategy can be used in LSRTM procedures of predicting the primary reflected data, calculating the step length, and computing the gradient. Theoretical analyses and numerical experiments are given to justify this computational strategy for LSRTM.

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28

Mursaleen, M., Shagufta Rahman, and Khursheed Ansari. "Approximation by Jakimovski-Leviatan-Stancu-Durrmeyer type operators." Filomat 33, no. 6 (2019): 1517–30. http://dx.doi.org/10.2298/fil1906517m.

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Анотація:

In the present paper, we introduce Stancu type modification of Jakimovski-Leviatan-Durrmeyer operators. First, we estimate moments of these operators. Next, we study the problem of simultaneous approximation by these operators. An upper bound for the approximation to rth derivative of a function by these operators is established. Furthermore, we obtain A-statistical approximation properties of these operators with the help of universal korovkin type statistical approximation theorem.

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29

QASIM, MOHD, M. MURSALEEN, ASIF KHAN, and ZAHEER ABBAS. "On some Statistical Approximation Properties of Generalized Lupas-Stancu Operators." Kragujevac Journal of Mathematics 46, no. 5 (2022): 797–813. http://dx.doi.org/10.46793/kgjmat2205.797q.

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Анотація:

The purpose of this paper is to introduce Stancu variant of generalized Lupaş operators whose construction depends on a continuously diﬀerentiable, increasing and unbounded function ρ. Depending on the selection of γ and δ, these operators are more ﬂexible than the generalized Lupaş operators while retaining their approximation properties. For these operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation. Finally, we investigate the statistical approximation property of the new operators with the aid of a Korovkin type statistical approximation theorem.

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30

Yingdian, Ma, and Wang Weimeng. "The Approximation of a Modified Baskakov Operator." Journal of Mathematics 2023 (March 1, 2023): 1–14. http://dx.doi.org/10.1155/2023/7767936.

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Анотація:

Because of their simple form and high quality, linear arithmetical operators, particularly linear positive operators, are very popular. The number of in-depth studies on linear operator approximation is extensive, and the majority of the published material falls into four categories: types and structures of operators, approximation of operators, order of approximation and the converse theorem of operators, and operator saturation. The Baskakov operator will be transformed in the same way to explore its approximation characteristics as well as the approximation theorem and converse theorem.

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31

Bede, Barnabás, Hajime Nobuhara, János Fodor, and Kaoru Hirota. "Max-Product Shepard Approximation Operators." Journal of Advanced Computational Intelligence and Intelligent Informatics 10, no. 4 (July 20, 2006): 494–97. http://dx.doi.org/10.20965/jaciii.2006.p0494.

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Анотація:

In crisp approximation theory the operations that are used are only the usual sum and product of reals. We propose the following problem: are sum and product the only operations that can be used in approximation theory? As an answer to this problem we propose max-product Shepard approximation operators and we prove that these operators have very similar properties to those provided by the crisp approximation theory. In this sense we obtain uniform approximation theorem of Weierstrass type, and Jackson-type error estimate in approximation by these operators.

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32

Goodman, T. N. T., and S. L. Lee. "Convolution operators with trigonometric spline kernels." Proceedings of the Edinburgh Mathematical Society 31, no. 2 (June 1988): 285–99. http://dx.doi.org/10.1017/s0013091500003412.

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Анотація:

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].

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33

Rempulska, Lucyna, and Mariola Skorupka. "Approximation by Kantorovich type operators." Acta et Commentationes Universitatis Tartuensis de Mathematica 7 (December 31, 2003): 57–70. http://dx.doi.org/10.12697/acutm.2003.07.06.

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Анотація:

We study approxiamtion properties of the Kantorovich type operators associated with the Szasz-Mirakyan and Baskakov operators. We prove that the approximation order of smoothness functions by considered operators is better than for the classical Szasz-Mirakyan and Baskakov operators given in [2]. Our paper is motivated by results obtained in [2], [6], [9] and [12].

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34

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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Анотація:

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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35

Zhang, Yan-Lan, and Chang-Qing Li. "Topological properties of a pair of relation-based approximation operators." Filomat 31, no. 19 (2017): 6175–83. http://dx.doi.org/10.2298/fil1719175z.

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Анотація:

Rough set theory is an important tool for data mining. Lower and upper approximation operators are two important basic concepts in the rough set theory. The classical Pawlak rough approximation operators are based on equivalence relations and have been extended to relation-based generalized rough approximation operators. This paper presents topological properties of a pair of relation-based generalized rough approximation operators. A topology is induced by the pair of generalized rough approximation operators from an inverse serial relation. Then, connectedness, countability, separation property and Lindel?f property of the topological space are discussed. The results are not only beneficial to obtain more properties of the pair of approximation operators, but also have theoretical and actual significance to general topology.

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36

Qiao, Sha, Ping Zhu, and Witold Pedrycz. "Rough set analysis of graphs." Filomat 36, no. 10 (2022): 3331–54. http://dx.doi.org/10.2298/fil2210331q.

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Анотація:

Relational data has become increasingly important in decision analysis in recent years, and so mining knowledge which preserves relationships between objects is an important topic. Graphs can represent the knowledge which contains objects and relationships between objects. Rough set theory provides an effective tool for extracting knowledge, but it is not sufficient to extract the knowledge containing the data on relationships between objects. In order to extend the application scope and enrich the rough set theory, it is essential to develop a rough set analysis of graphs. This extension is important because graphs play a crucial role in social network analysis. In this paper, the rough set analysis of graphs based on general binary relations is investigated. We introduce three types of approximation operators of graphs: vertex graph approximation operators, edge graph approximation operators, and graph approximation operators. Relationships between approximation operators of graphs and approximation operators of sets are presented. Then we investigate the approximation operators of graphs within constructive and axiomatic approaches.

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37

Gupta, Vijay, Ana Maria Acu, and Hari Mohan Srivastava. "Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives." Symmetry 12, no. 6 (June 2, 2020): 915. http://dx.doi.org/10.3390/sym12060915.

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In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupaş–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper.

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38

Gupta, Vijay. "Some Approximation Properties for Modified Baskakov Type Operators." gmj 12, no. 2 (June 2005): 217–28. http://dx.doi.org/10.1515/gmj.2005.217.

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Abstract We study some direct results for the recently introduced family of modified Baskakov type operators. In particular, we obtain local direct results on ordinary and simultaneous approximation and an estimation of error for linear combinations in terms of higher order modulus of continuity. We have applied the Steklov mean as a tool for the linear approximating method.

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39

Wang, Gang, and Hua Mao. "Approximation operators based on preconcepts." Open Mathematics 18, no. 1 (May 28, 2020): 400–416. http://dx.doi.org/10.1515/math-2020-0146.

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Abstract Using the notion of preconcept, we generalize Pawlak’s approximation operators from a one-dimensional space to a two-dimensional space in a formal context. In a formal context, we present two groups of approximation operators in a two-dimensional space: one is aided by an equivalence relation defined on the attribute set, and another is aided by the lattice theoretical property of the family of preconcepts. In addition, we analyze the properties of those approximation operators. All these results show that we can approximate all the subsets in a formal context assisted by the family of preconcepts using the above groups of approximation operators. Some biological examples show that the two groups of approximation operators provided in this article have potential ability to assist biologists to do the phylogenetic analysis of insects.

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40

Özarslan, Mehmet. "Approximation properties of Jain-Appell operators." Applicable Analysis and Discrete Mathematics, no. 00 (2020): 44. http://dx.doi.org/10.2298/aadm190223044o.

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In this paper, we introduce the Jain-Appell operators by applying Gamma transform to the Jakimovski-Leviatan operators. In their special cases they include not only the Jain-Pethe operators, but also new families of operators, where we call them Appell-Baskakov and Appell-Lupa? operators, since their special cases contain Baskakov and Lupa? operators, respectively. We investigate their weighted approximation properties and compute the error of approximation by using certain Lipschitz class functions. Furthermore, we obtain their A-statistical approximation property.

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41

Doğru, O., R. N. Mohapatra, and M. Örkcü. "Approximation properties of generalized Jain operators." Filomat 30, no. 9 (2016): 2359–66. http://dx.doi.org/10.2298/fil1609359d.

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Анотація:

In this paper, we investigate a variant of the Jain operators, which preserve the linear functions. We compute the rate of convergence of these operators with the help of K-functional. We also introduce modifications of the Jain operators based on the models in [4] and [10]. These modified operators yield better error estimates than the Jain operators.

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42

Cai, Qing-Bo, Asif Khan, Mohd Mansoori, Mohammad Iliyas та Khalid Khan. "Approximation by λ-Bernstein type operators on triangular domain". Filomat 37, № 6 (2023): 1941–58. http://dx.doi.org/10.2298/fil2306941c.

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In this paper, a new type of ?-Bernstein operators (Bwm,?g)(w,z) and (Bzn ?g)(w,z), their Products Pmn,?g (w,z), Qnm,?g (w, z), and their Boolean sums Smn,?g (w,z), Tnm,?g (w, z) are constructed on triangle h with parameter ? [1,1]. Convergence theorem for Lipschitz type continuous functions and a Voronovskaja-type asymptotic formula are studied for these operators. Remainder terms for error evaluation by using the modulus of continuity are discussed. Graphical representations are added to demonstrate the consistency of theoretical findings for the operators approximating functions on the triangular domain. Also, we show that the parameter ? will provide flexibility in approximation; in some cases, the approximation will be better than its classical analogue.

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43

Shvai, Olga, and Kateryna Pozharska. "On some approximation properties of Gauss-Weierstrass singular operators." Ukrainian Mathematical Bulletin 18, no. 4 (November 12, 2021): 560–68. http://dx.doi.org/10.37069/1810-3200-2021-18-4-7.

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Анотація:

Approximation theorems were formulated for function continuous in the neighborhood of some point $x$, $-\infty <x<\infty $. Namely, the upper bounds were obtained for the function approximations by their Gauss-Weierstrass singular operators in terms of a majorant function for the first- and second-order continuity moduli of the relevant functions.

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44

Gupta, Vijay, and G. Srivastava. "Approximation by Durrmeyer-type operators." Annales Polonici Mathematici 64, no. 2 (1996): 153–59. http://dx.doi.org/10.4064/ap-64-2-153-159.

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45

FINTA, ZOLTAN. "Approximation by q-parametric operators." Publicationes Mathematicae Debrecen 78, no. 3-4 (April 1, 2011): 543–56. http://dx.doi.org/10.5486/pmd.2011.4733.

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46

Cobos, Fernando, Oscar Domínguez, and Antón Martínez. "Compact operators and approximation spaces." Colloquium Mathematicum 136, no. 1 (2014): 1–11. http://dx.doi.org/10.4064/cm136-1-1.

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47

Anastassiou, George A. "APPROXIMATION BY DISCRETE SINGULAR OPERATORS." Cubo (Temuco) 15, no. 1 (March 2013): 97–112. http://dx.doi.org/10.4067/s0719-06462013000100006.

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48

Yang, X. R. "Approximation of Generalized Bernstein Operators." Analysis in Theory and Applications 30, no. 2 (June 2014): 205–13. http://dx.doi.org/10.4208/ata.2014.v30.n2.6.

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49

Gupta, Vijay, and P. N. Agrawal. "Approximation by modified Păltănea operators." Publications de l'Institut Math?matique (Belgrade) 107, no. 121 (2020): 157–64. http://dx.doi.org/10.2298/pim2021157g.

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Анотація:

We discuss some approximation properties of hybrid genuine operators. We find central moments using the concept of moment generating function. A quantitative Voronovskaya and Gruss-Voronovskaya type theorem are proven. Also, we obtain the degree of approximation of the considered operators by means of the second order Ditzian-Totik modulus of smoothness.

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50

Grabowski, Adam, and Michał Sielwiesiuk. "Formalizing Two Generalized Approximation Operators." Formalized Mathematics 26, no. 2 (July 1, 2018): 183–91. http://dx.doi.org/10.2478/forma-2018-0016.

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Анотація:

Summary Rough sets, developed by Pawlak [15], are important tool to describe situation of incomplete or partially unknown information. In this article we give the formal characterization of two closely related rough approximations, along the lines proposed in a paper by Gomolińska [2]. We continue the formalization of rough sets in Mizar [1] started in [6].

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