On Szász-Mirakyan-Jain Operators preserving exponential functions

In this article, we establish the approximation by Durrmeyer type Jakimovski–Leviatan operators involving the Brenke type polynomials. The positive linear operators are constructed for the Brenke polynomials, and thus approximation properties for these polynomials are obtained. The order of convergence and the weighted approximation are also considered. Finally, the Voronovskaya type theorem is demonstrated for some particular case of these polynomials.

In this research paper, we construct a new sequence of Riemann–Liouville type fractional ‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem.

By using given arbitrary sequences,property that limn 1nn= 0and limn 1 n= 0, we give a Kantorovichtype generalization of Jain operator based on the a Poisson disrtibition. Fristlywe give the quantitative Voronovskaya type theorem. Then we also obtain theGrüss Voronovskaya type theorem in quantitative form .We show that theyhave an arbitrary good order of weighted approximation

The goal of this paper is to propose a modification of Lupaş-Jain operators based on a function ρ having some properties. Primarily, the convergence of given operators in weighted spaces is discussed. Then, order of approximation via weighted modulus of continuity is computed for these operators. Further, Voronovskaya type theorem in quantitative form is taken into consideration, as well. Ultimately, some graphical results that illustrate the convergence of investigated operators to f are given.

In this work, we obtain the approximation properties of a new generalization of Szász-Mirakjan operators based on postquantum calculus.Firstly, for these operators, a recurrence formulation for the moments is obtained, and up to the fourth degree, the central moments are examined.Then, a local approximation result is attained.Furthermore, the degree of approximation in respect of the modulus of continuity on a finite closed set and the class of Lipschitz are computed.Next, the weighted uniform approximation on an unbounded interval is showed, and by the modulus of continuity, the order of convergence is estimated.Lastly, we proved the Voronovskaya type theorem and gave some illustrations to compare the related operators' convergence to a certain function.

Stancu generalized Baskakov operators using inverse Polya–Eggenberger distribution for a real valued bounded function on $$[0,\infty )$$ and a non-negative real number $$\alpha $$ (which may depend on n). Dhamija and Deo (Appl Math Comput 286:15–22, 2016) introduced a Durrmeyer type modification of these generalized operators and studied the uniform convergence, the rate of convergence by means of the moduli of continuity and the Peetre’s K-functional. The purpose of the present paper is to continue to study the approximation properties of these Durrmeyer type operators. Local and global direct approximation theorems and a Voronovskaya type asymptotic theorem are established. The quantitative Voronovskaya and Gruss Voronovskaya type theorems are investigated by calculating the indispensible sixth order moment. The weighted approximation properties and the approximation of functions with derivatives of bounded variation are also studied.

In this manuscript, we construct new modification of Baskakov operators on (0,?) using the second central moment of the classical Baskakov operators. And the moments and the central moments computation formulas and their quantitative properties are computed. Then, rate of convergence, point-wise estimates, weighted approximation and Voronovskaya type theorem for the new operators are established. Also, Kantorovich and Durrmeyer type generalizations are discussed. Finally, some graphs and numerical examples are showed by using Matlab algorithms.

<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>

Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, ${\mathit{e}}^{2}$${\mathbf{A}}^{2}$/2${\mathit{mc}}^{2}$, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency.An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.

In this paper, we construct the Chlodowsky-type Szász operators defined via Boas–Buck-type polynomials. We prove some approximation properties and obtain the rate of the convergence for these operators. We also study the Voronovskaya-type theorem and weighted approximation.

A generalization of Szász–Mirakyan operators based on [formula omitted]-integers

In this In this paper, we aim to obtain several approximation properties of Szasz-Mirakjan-Baskakov operators with shape parameter lambda in [-1,1]. We reach some preliminary results such as moments and central moments. Next, we estimate the order of convergence with respect to the usual modulus of continuity, for the functions belong to Lipschitz-type class and Peetre's K-functional, respectively. Also, we prove a result concerning the weighted approximation for these operators. Finally, we give the comparison of the convergence of these newly defined operators to certain functions with some graphics.

Based on the canonical correlation analysis, we derive series representations of the probability density function (PDF) and the cumulative distribution function (CDF) of the information density of arbitrary Gaussian random vectors as well as a general formula to calculate the central moments. Using the general results, we give closed-form expressions of the PDF and CDF and explicit formulas of the central moments for important special cases. Furthermore, we derive recurrence formulas and tight approximations of the general series representations, which allow efficient numerical calculations with an arbitrarily high accuracy as demonstrated with an implementation in Python publicly available on GitLab. Finally, we discuss the (in)validity of Gaussian approximations of the information density.

It is known that the rotation, scaling and translation invariant property of image moments has a high significance in image recognition. For this reason, the seven invariant moments presented by Hu are widely used in the field of image analysis. These moments are of finite order; therefore, they do not comprise a complete set of image descriptors. For this reason, we introduce in this paper another series of invariant moments of infinite order, which are based on normalized central moments. The non-orthogonal property of these moments causes the redundancy of information. To overcome this problem, we propose a new construction technique of non-separable orthogonal polynomials in two variables based on a recurrence formula and we present a new set of orthogonal moments, which are invariant to translation, scaling and rotation. The presented approaches are tested in several well-known computer vision datasets including moment’s invariability, image retrieval and classification of objects, this latter based on fuzzy K-means clustering algorithm. The performance of these invariant moments for classification and image retrieval is compared with some recent invariant moments such as invariants of multi-channel orthogonal radial-substituted Chebyshev moments, invariants of quaternion radial-substituted Chebyshev moments, invariants of rotational moments in Radon space and Legendre–Fourier moments in Radon space. The experimental results made using four databases of images, namely Columbia Object Image Library (COIL-20) database, MPEG7-CE shape database, COIL-100 database and ORL database, show that our orthogonal invariant moments have done better than the other descriptors tested.

About a system of anti-periodic trigonometric functions