scholarly journals Approximation by Generalized Lupaş Operators Based on q-Integers

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 68 ◽  
Author(s):  
Mohd Qasim ◽  
M. Mursaleen ◽  
Asif Khan ◽  
Zaheer Abbas
Keyword(s):  
Local Approximation ◽  
Approximation Properties ◽  
Unbounded Function ◽  
Quantitative Estimates ◽  
Lupaş Operators ◽  
Selection Of

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.

scholarly journals Convergence of modified Szász-Mirakyan-Durrmeyer operators depending on certain parameters

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohd Qasim ◽  
Mohd Shanawaz Mansoori ◽  
Asif Khan ◽  
Zaheer Abbas ◽  
Mohammad Mursaleen
Keyword(s):  
Type Theorem ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Approximation Properties ◽  
Unbounded Function ◽  
Durrmeyer Operators ◽  
Quantitative Estimates ◽  
Voronovskaya Type Theorem ◽  
Selection Of

<p style='text-indent:20px;'>Motivated by certain generalizations, in this paper we consider a new analogue of modified Szá sz-Mirakyan-Durrmeyer operators whose construction depends on a continuously differentiable, increasing and unbounded function <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> with extra parameters <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>. Depending on the selection of <inline-formula><tex-math id="M4">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, these operators are more flexible than the modified Szá sz-Mirakyan-Durrmeyer operators while retaining their approximation properties. For these operators we give weighted approximation, Voronovskaya type theorem and quantitative estimates for the local approximation.</p>

scholarly journals A new construction of Lupaş operators and its approximation properties

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohd Qasim ◽  
Asif Khan ◽  
Zaheer Abbas ◽  
Qing-Bo Cai
Keyword(s):  
Uniform Approximation ◽  
Type Theorem ◽  
Local Approximation ◽  
Moduli Of Smoothness ◽  
Approximation Properties ◽  
Quantitative Estimates ◽  
Voronovskaya Type Theorem ◽  
Lupaş Operators ◽  
New Construction ◽  
Weighted Moduli Of Smoothness

AbstractThe aim of this paper is to study a new generalization of Lupaş-type operators whose construction depends on a real-valued function ρ by using two sequences $u_{m} $ u m and $v_{m}$ v m of functions. We prove that the new operators provide better weighted uniform approximation over $[0,\infty )$ [ 0 , ∞ ) . In terms of weighted moduli of smoothness, we obtain degrees of approximation associated with the function ρ. Also, we prove Voronovskaya-type theorem, quantitative estimates for the local approximation.

scholarly journals Convergence properties of generalized Lupaş-Kantorovich operators

2021 ◽  
Vol 13 (3) ◽  
pp. 818-830
Author(s):  
M. Qasim ◽  
A. Khan ◽  
Z. Abbas ◽  
M. Mursaleen
Keyword(s):  
Type Theorem ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Convergence Properties ◽  
Unbounded Function ◽  
Quantitative Estimates ◽  
Voronovskaya Type Theorem ◽  
Lupaş Operators ◽  
Kantorovich Operators

In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $\rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.

scholarly journals Convergence of Generalized Lupaş-Durrmeyer Operators

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 852 ◽  
Author(s):  
Mohd Qasim ◽  
Mohammad Mursaleen ◽  
Asif Khan ◽  
Zaheer Abbas
Keyword(s):  
Weighted Approximation ◽  
Local Approximation ◽  
Basis Functions ◽  
Integral Type ◽  
Central Moments ◽  
Durrmeyer Operators ◽  
Quantitative Estimates ◽  
Lupaş Operators

The main aim of this paper is to establish summation-integral type generalized Lupaş operators with weights of Beta basis functions which depends on μ having some properties. Primarily, for these new operators, we calculate moments and central moments, weighted approximation is discussed. Further, Voronovskaya type asymptotic theorem is proved. Finally, quantitative estimates for the local approximation is taken into consideration.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

scholarly journals Generalized p,q-Gamma-type operators

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Wen-Tao Cheng ◽  
Qing-Bo Cai
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Type Theorem ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Pointwise Estimates ◽  
Approximation Properties ◽  
Central Moments ◽  
Voronovskaja Type Theorem

In the present paper, the generalized p,q-gamma-type operators based on p,q-calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K-functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.

scholarly journals A nonconforming Trefftz virtual element method for the Helmholtz problem

2019 ◽  
Vol 29 (09) ◽  
pp. 1619-1656 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Convergence Rates ◽  
Plane Waves ◽  
Local Approximation ◽  
Approximation Properties ◽  
Virtual Element Method ◽  
Impedance Boundary ◽  
Helmholtz Problem ◽  
Text Version ◽  
Virtual Element ◽  
Element Method

We introduce a novel virtual element method (VEM) for the two-dimensional Helmholtz problem endowed with impedance boundary conditions. Local approximation spaces consist of Trefftz functions, i.e. functions belonging to the kernel of the Helmholtz operator. The global trial and test spaces are not fully discontinuous, but rather interelement continuity is imposed in a nonconforming fashion. Although their functions are only implicitly defined, as typical of the VEM framework, they contain discontinuous subspaces made of functions known in closed form and with good approximation properties (plane-waves, in our case). We carry out an abstract error analysis of the method, and derive [Formula: see text]-version error estimates. Moreover, we initiate its numerical investigation by presenting a first test, which demonstrates the theoretical convergence rates.

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