scholarly journals Phillips-Type q -Bernstein Operators on Triangles

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Asif Khan ◽  
M. S. Mansoori ◽  
Khalid Khan ◽  
M. Mursaleen
Keyword(s):  
Modulus Of Continuity ◽  
Classical Case ◽  
Approximation Formula ◽  
Graphical Representations ◽  
Bernstein Operators ◽  
Quantum Analogue ◽  
Boolean Sums ◽  
Peano’S Theorem

The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operators B m , q u f u , v and B n , q v f u , v , their products P m n , q f u , v and Q n m , q f u , v , their Boolean sums S m n , q f u , v and T n m , q f u , v on triangle T h , which interpolate a given function on the edges, respectively, at the vertices of triangle using quantum analogue. Based on Peano’s theorem and using modulus of continuity, the remainders of the approximation formula of corresponding operators are evaluated. Graphical representations are added to demonstrate consistency to theoretical findings. It has been shown that parameter q provides flexibility for approximation and reduces to its classical case for q = 1 .

scholarly journals Lupaş type Bernstein operators on triangles based on quantum analogue

2021 ◽  
Vol 60 (6) ◽  
pp. 5909-5919
Author(s):  
Asif Khan ◽  
M.S. Mansoori ◽  
Khalid Khan ◽  
M. Mursaleen
Keyword(s):  
Bernstein Operators ◽  
Quantum Analogue

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

scholarly journals ON THE NATURE OF COHERENTS IN THE SYSTEM OF OSCILLATORS

2019 ◽  
pp. 91-95
Author(s):  
V.M. Kuklin
Keyword(s):  
Spatial Variation ◽  
Quantum System ◽  
Field Intensity ◽  
Radiation Intensity ◽  
Population Inversion ◽  
Classical Case ◽  
Radiation Energy ◽  
Dispersion Characteristics ◽  
Quantum Analogue ◽  
Integral Field

The paper presents the transition to the regime of induced radiation of a system of oscillators in the classical and the quantum cases. This transition occurs due to synchronization by the integral field of the phases of a small part of oscillator-emitters. In the quantum analogue of this model, it is shown that the formation of an induced (and, therefore, coherent, as noted by Ch. Towns) pulse of the field is due to the interference of nutation of population inversion in different regions of the system of oscillators. The law of spatial variation of the field intensity is deter-mined by the dispersion characteristics of the system and the level of absorption or output of the radiation energy. Only a small fraction of oscillators provide induced radiation: 8% in the classical case and half as much in the case of a quantum system, where a change in the sign of population inversion in the regions of the highest field values significantly affects the limitation of the radiation intensity.

On Simultaneous Approximation by Iterated Boolean Sums of Bernstein Operators

2014 ◽  
Vol 66 (1-2) ◽  
pp. 21-41 ◽  
Author(s):  
Borislav R. Draganov
Keyword(s):  
Simultaneous Approximation ◽  
Bernstein Operators ◽  
Boolean Sums

scholarly journals Approximation properties of the new type generalized Bernstein-Kantorovich operators

2021 ◽  
Vol 7 (3) ◽  
pp. 3826-3844
Author(s):  
Mustafa Kara ◽  
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Lipschitz Function ◽  
Type Theorem ◽  
Numerical Results ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
Voronovskaya Type Theorem ◽  
New Type ◽  
Kantorovich Operators

<abstract><p>In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.</p></abstract>

scholarly journals FRACTAL APPROXIMATION OF JACKSON TYPE FOR PERIODIC PHENOMENA

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850079 ◽  
Author(s):  
M. A. NAVASCUÉS ◽  
SANGITA JHA ◽  
A. K. B. CHAND ◽  
M. V. SEBASTIÁN
Keyword(s):  
Functional Model ◽  
Classical Case ◽  
Sampling Frequency ◽  
Trigonometric Polynomials ◽  
Approximation Formula ◽  
Fractal Interpolation ◽  
Trigonometric Functions ◽  
Fractal Functions ◽  
Fractal Approximation ◽  
Range Of Values

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators and their classical counterparts are proposed.

scholarly journals Lagrange-type operators associated with Uan

2014 ◽  
Vol 96 (110) ◽  
pp. 159-168 ◽  
Author(s):  
Heiner Gonska ◽  
Ioan Raşa ◽  
Elena-Dorina Stănilă
Keyword(s):  
Main Tool ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operators ◽  
Boolean Sums ◽  
Derivatives Of

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.

scholarly journals Approximation theorems for the iterated Boolean sums of Bernstein operators

1994 ◽  
Vol 53 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Heinz H. Gonska ◽  
Xin-long Zhou
Keyword(s):  
Bernstein Operators ◽  
Approximation Theorems ◽  
Boolean Sums

scholarly journals On Approximation Properties of (p,q)-Bernstein Operators

2018 ◽  
Vol 11 (2) ◽  
pp. 457-467
Author(s):  
Döne Karahan ◽  
Aydın İzgi
Keyword(s):  
Modulus Of Continuity ◽  
Bernstein Operators ◽  
Approximation Properties

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.

scholarly journals Strong inequalities for the iterated Boolean sums of Bernstein operators

2019 ◽  
Vol 64 (3) ◽  
pp. 299-304
Author(s):  
Li Cheng ◽  
◽  
Xinlong Zhou ◽  
◽  
Keyword(s):  
Bernstein Operators ◽  
Boolean Sums
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