The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators. r 2004 Elsevier Inc. All rights reserved.

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.

This paper introduced a family of neural networks multivariate square rational Bernstein operators defined by extending the artificial using polynomials and studied behavior this network. Also, gave application through some numerical examples.