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

V. B. Kiran Kumar, P. C. Vinaya

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.

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

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.
Paper Structure (7 sections, 5 theorems, 38 equations)

This paper contains 7 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The Main Result
  3. Examples
  4. Kantorovich-type operators
  5. Bernstein-type operators
  6. Trigonometric analogue and Applications
  7. Application to Preconditioners of Toeplitz systems

Key Result

Theorem 1.3

shisha Let $\{L_n\}\_{n\in\mathbb{N}}$ be a sequence of positive linear operators with the same domain $D$ which contains the restrictions of $1,t,t^2$ to $[a,b]\subseteq \mathbb{R}$. For $n=1,2,\ldots$, suppose that $L_n(1)$ is bounded. Let $f\in D$ be continuous in $[a,b]$, and $\|.\|$ the sup-nor In particular, if $L_n(1)=1$, this inequality becomes $\|f-L_n(f)\|\leq 2\omega(f,\mu_n)$.

Theorems & Definitions (23)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['mains']}
  • Remark 2.3
  • Example 3.1
  • Example 3.2
  • ...and 13 more