Bernstein–Jacobi-type operators preserving derivatives

• Published in: Computational & applied mathematics Vol. 43; no. 5
• Main Authors: Lara-Velasco, David, Pérez, Teresa E.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-07-2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Eigenvectors; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Operators (mathematics); Polynomials; Primary 33C50; Classical Jacobi polynomials; Extended Jacobi polynomials; Bernstein-type operators; 42C05
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-024-02796-2

A general frame for Bernstein-type operators that preserve derivatives is given. We introduce Bernstein-type operators based in the weighted classical Jacobi inner product on the interval [0, 1] that extend the well known Bernstein–Durrmeyer operator as well as some other types of Bernstein operators that appear in the literature. Apart from standard results, we deduce properties about the preservation of derivatives and prove that classical Jacobi orthogonal polynomials on [0, 1] are the eigenfunctions of these operators. We also study the limit cases when one of the parameters of the Jacobi polynomials is a negative integer. Finally, we study several numerical examples.