Weighted A-statistical convergence for sequences of positive linear operators

Area of Science:

• Mathematical Analysis
• Approximation Theory
• Sequence Spaces

Background:

• A-statistical convergence is a significant tool in summability theory.
• The Korovkin approximation theorem is fundamental in establishing convergence of sequences of linear operators.
• Weighted convergence methods offer enhanced analytical capabilities.

Purpose of the Study:

• To introduce and define weighted A-statistical convergence for sequences.
• To extend the Korovkin approximation theorem using this new convergence concept.
• To analyze the rate of this convergence and provide a practical illustration.

Main Methods:

• Definition of weighted A-statistical convergence using nonnegative regular matrices.
• Application of the new convergence concept to prove the Korovkin approximation theorem.
• Derivation of the rate of weighted A-statistical convergence.
• Construction of an example using Bernstein polynomials.

Main Results:

• The successful introduction of weighted A-statistical convergence.
• A novel proof of the Korovkin approximation theorem based on weighted A-statistical convergence.
• Quantification of the rate of weighted A-statistical convergence.
• An illustrative example validating the theoretical findings.

Conclusions:

• Weighted A-statistical convergence provides a refined framework for sequence analysis.
• The study successfully extends approximation theory results.
• The established rate of convergence offers practical insights for applications.