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Generalized Jacobi-Weierstrass operators and Jacobi expansions.
José A Adell1, Jorge Bustamante2, Juan J Merino3
11Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, Zaragoza, Spain.
Journal of Inequalities and Applications
|August 24, 2018
Summary
This study introduces generalized Jacobi-Weierstrass operators for K-functionals in Jacobi expansions. It also explores fractional operator powers and approximation inequalities.
Area of Science:
- Approximation Theory
- Harmonic Analysis
- Functional Analysis
Background:
- Jacobi expansions are fundamental in approximation theory.
- K-functionals measure the smoothness of functions.
- Generalized Jacobi-Weierstrass operators offer a new framework for analysis.
Purpose of the Study:
- To realize K-functionals using generalized Jacobi-Weierstrass operators.
- To investigate fractional powers of these operators.
- To establish simultaneous approximation and Nikolskii-Stechkin inequalities.
Main Methods:
- Construction of generalized Jacobi-Weierstrass operators.
- Application of operator theory to K-functionals.
- Analysis of approximation properties and inequalities.
Main Results:
- A concrete realization of K-functionals is provided.
- Properties of fractional powers of the operators are derived.
- New inequalities of Nikolskii-Stechkin type are established.
Conclusions:
- The generalized Jacobi-Weierstrass operators offer a powerful tool for studying K-functionals.
- The results extend existing approximation theory findings.
- Further research can explore applications in other areas of analysis.
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