MODIFIED BERNSTEIN POLYNOMIALS AND JACOBI POLYNOMIALS IN Q-CALCULUS

By M. Derriennic. 2004

80 citations recorded by the source corpus.

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the $q$-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in $]0,1[$. Numerous properties of the modified Bernstein Polynomials are extended to their $q$-analogues: simultaneous approximation, pointwise convergence even for unbounded functions, shape-preserving property, Voronovskaya theorem, self-adjointness. Some properties of the eigenvectors, which are $q$-extensions of Jacobi polynomials, are given.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a94cd5dfb533a404cea21ec7417cfc87b81eaa61

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In Applied Mathematics

1. Carmen Violeta Muraru
2. Cristina Radu
3. Vijay Gupta
4. Tuncer Acar

In Numerical Analysis

1. Carmen Violeta Muraru
2. Cristina Radu
3. Vijay Gupta
4. Tuncer Acar

In Statistics and Probability

1. Carmen Violeta Muraru
2. Cristina Radu
3. Vijay Gupta
4. Tuncer Acar
5. Ali Aral

Citations in the corpus

• Statistical approximation properties of q-Baskakov-Kantorovich operators (2009, methodology): “Recently, generalizations of Durrmeyer and Kantorovich operators based on q-integers were studied in [ 6 ],[13] and [18].”
• Note on q-Bernstein-Schurer operators (2011, methodology): “Durrmeyer type generalization of the operators based on q-integers was studied by Derriennic in [6].”