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 .
<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>
In this paper, we extend the properties of rational Lupa?-Bernstein blending functions, Lupa?-B?zier curves and surfaces over arbitrary compact intervals [?,?] in the frame of post quantum-calculus and derive the de-Casteljau?s algorithm based on post quantum-integers. We construct a two parameter family as Lupa? post quantum Bernstein functions over arbitrary compact intervals and establish their degree elevation and reduction properties. We also discuss some fundamental properties over arbitrary intervals for these curves such as de Casteljau algorithm and degree evaluation properties. Further we construct post quantum Lupa? Bernstein operators over arbitrary compact intervals with the help of rational Lupa?- Bernstein functions. At the end some graphical representations are added to demonstrate consistency of theoretical findings.
This paper deals with Lupaş post quantum Bernstein operators over arbitrary closed and bounded interval constructed with the help of Lupaş post quantum Bernstein bases. Due to the property that these bases are scale invariant and translation invariant, the derived results on arbitrary intervals are important from computational point of view. Approximation properties of Lupaş post quantum Bernstein operators on arbitrary compact intervals based on Korovkin type theorem are studied. More general situation along all possible cases have been discussed favouring convergence of sequence of Lupaş post quantum Bernstein operators to any continuous function defined on compact interval. Rate of convergence by modulus of continuity and functions of Lipschitz class are computed. Graphical analysis has been presented with the help of MATLAB to demonstrate approximation of continuous functions by Lupaş post quantum Bernstein operators on different compact intervals.
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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