Genuine modified Bernstein-Durrmeyer operators.

Area of Science:

• Approximation Theory
• Numerical Analysis
• Functional Analysis

Background:

• Bernstein-Durrmeyer operators are fundamental in approximation theory.
• Understanding the convergence properties of these operators is crucial for numerical applications.
• Previous research has focused on various aspects of operator approximation, but novel variants warrant further investigation.

Purpose of the Study:

• To introduce and analyze genuine Bernstein-Durrmeyer operators.
• To investigate the rate of convergence using the Peetre [Formula: see text]-functional and modulus of smoothness.
• To establish quantitative Voronovskaya type and Grüss-Voronovskaya type theorems.

Main Methods:

• Utilizing the Peetre [Formula: see text]-functional and modulus of smoothness for convergence analysis.
• Applying quantitative Voronovskaya type theorem to assess local approximation behavior.
• Employing Grüss-Voronovskaya type theorem for quantitative mean convergence analysis.

Main Results:

• The paper presents novel Bernstein-Durrmeyer operators with specific function-preserving properties.
• Quantitative rates of convergence are established using advanced approximation tools.
• Theorems of Voronovskaya and Grüss-Voronovskaya types are derived in a quantitative mean sense.

Conclusions:

• The newly developed Bernstein-Durrmeyer operators offer a valuable addition to approximation theory.
• The established convergence rates and theorems provide a robust theoretical foundation.
• Graphical representations illustrate the operators' behavior and effectiveness.