where λ1 ≥ · · · ≥ λn > 0 are the eigenvalues of A. It is a very useful tool to study the inefficiency of the ordinary least-squares estimate with one regressor in the linear model. Watson 1 introduced the ratio of the variance of the best linear unbiased estimator to the variance of the ordinary least-squares estimator. Such a lower bound of this ratio was provided by Kantorovich inequality 1....

In this paper, we apply four-dimensional infinite matrices to newly constructed original extension of bivariate Bernstein-Kantorovich type operators based on multiple shape parameters. We also use B?gel continuity construct the GBS (Generalized Boolean Sum) for defined Kantorovich type. Moreover, demonstrate certain illustrative graphs show applicability and validity proposed operators.

Our main purpose of this article is to study the convergence and other related properties q-Bernstein-Kantorovich operators including shifted knots real positive numbers. We design Bernstein-Kantorovich generated by basic q-calculus. More precisely, we our new in space continuous functions Lebesgue space. obtain degree with help modulus continuity integral continuity. Furthermore, establish qua...

The problem of optimal transportation between a set of sources and a set of wells has become recently the object of new mathematical models generalizing the Monge-Kantorovich problem. These models are more realistic as they predict the observed branching structure of communication networks. They also define new distances between measures. The question arises of how these distances compare to th...

In this paper, we deal with Stancu operators which depend on a non-negative integer parameter. Firstly, define Kantorovich extension of the operators. For functions belonging to space Lp [0, 1] , 1 ? p < ?, obtain convergence in norm by sequence Stancu-Kantorovich operators, and give an estimate for rate via first order averaged modulus smoothness. Moreover, operators; search variation detra...

The aim of this paper is to study the convergence and approximation properties of the bivariate operators and GBS operators of Kantorovich type.