On multivariate Lupa 3 operators
This paper is primarily concerned with multivariate Lupa3 operator. We demonstrate that multivariate Lupa3 operator preserves the properties of the general function of modulus of continuity, Lipschitz's constant and order of a Lipschitz continuous function. Moreover, we obtain monotonicity of the multivariate Lupa3 operator under the condition that the original function is convex. Lastly, two modi ed extensions are constructed.
On multivariate Baskakov operator
Abstract In this paper, we study multivariate Baskakov operator B n , d ( f , x ) . We first show that the operator can retain some properties of the original function f , such as monotony, semi-additivity and Lipschitz condition, etc. Secondly, we discuss the monotony on the sequence of multivariate Baskakov operator B n , d ( f , x ) for n when the function f is convex. Then, we propose, for estimating the rate of approximation, a new modulus of smoothness and prove the modulus to be equivalent to certain K -functional. Finally, with the modulus of smoothness as metric, we establish a strong direct theorem by using a decomposition technique for the operator.
Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces
We study the existence and shape preserving properties of a generalized Bernstein operator Bn fixing a strictly positive function f0, and a second function f1 such that f1/f0 is strictly increasing, within the framework of extended Chebyshev spaces Un. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator Bn : C[a, b] → Un with strictly increasing nodes, fixing $${f_{0}, f_{1} \in U_{n}}$$ . If $${U_{n} \subset U_{n + 1}}$$ and Un+1 has a non-negative Bernstein basis, then there exists a Bernstein operator Bn+1 : C[a, b] → Un+1 with strictly increasing nodes, fixing f0 and f1. In particular, if f0, f1, . . . , fn is a basis of Un such that the linear span of f0, . . . , fk is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator Bn with increasing nodes fixing f0 and f1. The second main result says that under the above assumptions the following inequalities hold$$B_{n} f \geq B_{n+1} f \geq f$$for all (f0, f1)-convex functions $${f \in C \, [ a, b]}$$ . Furthermore, Bnf is (f0, f1)-convex for all (f0, f1)-convex functions $${f \in C \, [a, b]}$$ .