Operator version of Korovkin Theorem; Degree of Convergence and its Applications

TL;DR

This paper extends the operator version of the Korovkin theorem by providing quantitative estimates of convergence, exploring applications including trigonometric cases and preconditioning large Toeplitz systems.

Contribution

It offers a quantitative estimate for Dumitru Popa's operator Korovkin theorem and applies it to various examples and the Toeplitz system preconditioning problem.

Findings

Quantitative convergence estimates for operator Korovkin theorem

Application to trigonometric analogue of the theorem

Use in preconditioning large Toeplitz systems

Abstract

In a recent article, Dumitru Popa proved an operator version of the Korovkin theorem. We recall the quantitative version of the Korovkin theorem obtained by O. Shisha and B. Mond in 1968. In this paper, we obtain a quantitative estimate for the operator version of the Korovkin theorem obtained by Dumitru Popa. We also consider various examples where the operator version is applicable and obtain similar estimates leading to the degree of convergence. In addition, we obtain the trigonometric analogue of this result by proving the quantitative version. Finally, we apply this result to the preconditioning problem of large linear systems with the Toeplitz structure.