This paper deals with weighted Lupaş post-quantum Bernstein blending functions and Bézier curves constructed with the help of bases via ( p , q ) \left(p,q) -integers. These blending functions form normalized totally positive bases. Due to the rational nature of weighted Lupaş post-quantum Bézier curves and positive weights, they help in investigating from geometric point of view. Their degree elevation properties and de Casteljau algorithm have been studied. It has been shown that quadratic weighted Lupaş post-quantum Bézier curves can represent conic sections in two-dimensional plane. Graphical analysis has been presented to discuss geometric interpretation of weight and conic section representation by weighted Lupaş post-quantum Bézier curves. This new generalized weighted Lupaş post-quantum Bézier curve provides better approximation and flexibility to a particular control point as well as control polygon due to extra parameter p p and q q in comparison to classical rational Bézier curves, Lupaş q q -Bézier curves and weighted Lupaş q q -Bézier curves.
The aim of this article is to construct univariate Bernstein-type operators ( ℬ m x G ) ( x , z ) \left({{\mathcal{ {\mathcal B} }}}_{m}^{x}G)\left(x,z) and ( ℬ n z G ) ( x , z ) , \left({{\mathcal{ {\mathcal B} }}}_{n}^{z}G)\left(x,z), their products ( P m n G ) ( x , z ) \left({{\mathcal{P}}}_{mn}G)\left(x,z) , ( Q n m G ) ( x , z ) \left({{\mathcal{Q}}}_{nm}G)\left(x,z) , and their Boolean sums ( S m n G ) ( x , z ) \left({{\mathcal{S}}}_{mn}G)\left(x,z) , ( T n m G ) ( x , z ) \left({{\mathcal{T}}}_{nm}G)\left(x,z) on elliptic region, which interpolate the given real valued function G G defined on elliptic region on its boundary. The bound of the remainders of each approximation formula of corresponding operators are computed with the help of Peano’s theorem and modulus of continuity, and the rate of convergence for functions of Lipschitz class is computed.
In this paper, Lupa? Bernstein-Kantorovich operators have been studied using Jackson and Riemann type (p,q)-integrals. It has been shown that (p, q)-integrals as well as Riemann type (p, q)-integrals are not well defined for 0 < q < p < 1 and thus further analysis is needed. Throughout the paper, the case 1 ? q < p < ? has been used. Advantages of using Riemann type (p, q)-integrals are discussed over general (p, q)-integrals. Lupa? Bernstein-Kantorovich operators constructed via Jackson integral need not be positive for every f ? 0. So to make these operators based on general (p, q)-integral positive, one need to consider strictly monotonically increasing functions, and to handle this situation Lupa? Bernstein-Kantorovich operators are constructed using Riemann type (p, q)-integrals. However Lupa? (p, q)-Bernstein-Kantorovich operators based on Riemann type (p, q)-integrals are always positive linear operators. Approximation properties for these operators based on Korovkin?s type approximation theorem are investigated. The rate of convergence via modulus of continuity and function f belonging to the Lipschitz class is computed.
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.
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