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Generalized Jacobi–Koornwinder’s-type Bernstein polynomials bases transformations

Mohammad A. AlQudah

School of Basic Sciences and Humanities, German Jordanian University, Amman 11180, Jordan

E-mail Address: mohammad.qudah@gju.edu.jo

Corresponding author.

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and

Maalee N. AlMheidat

Department of Basic Sciences, University of Petra, Amman 11196, Jordan

E-mail Address: malmheidat@uop.edu.jo

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https://doi.org/10.1142/S0129167X16500865Cited by:3 (Source: Crossref)

Abstract

This paper provides an explicit closed form of generalized Jacobi–Koornwinder’s polynomials of degree $r \leq n$ in terms of the Bernstein basis of fixed degree $n .$ Moreover, explicit forms of generalized Jacobi–Koornwinder’s type and Bernstein polynomials bases transformations are considered.

Keywords:

AMSC: 33C45, 33C50, 42C05, 05A10, 33B15

References

1. M. A. AlQudah, Generalized Tschebyscheff of the second kind and Bernstein polynomials change of bases, Eur. J. Pure Appl. Math. 8(3) (2015) 324–331. Web of Science, Google Scholar
2. H. Bavicnk and R. Koekoek, On a difference equation for generalizations of Charlier polynomials, J. Approx. Theory 81 (1995) 195–206. Crossref, Web of Science, Google Scholar
3. G. Farin, Curves and Surface for Computer Aided Geometric Design: A Practical Guide, Computer Science and Scientific Computing Series (Academic Press, MA, 1993). Google Scholar
4. R. T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Comput.-Aided Geom. Des. 29(6) (2012) 379–419. Crossref, Web of Science, Google Scholar
5. R. T. Farouki and V. T. Rajan, Algorithms for polynomials in Bernstein form, Comput.-Aided Geom. Des. 5(1) (1988) 1–26. Crossref, Google Scholar
6. I. Gradshtein and I. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1980). Google Scholar
7. J. Hoschek, D. Lasser and L. L. Schumaker, Fundamentals of Computer Aided Geometric Design (A. K. Peters, 1993). Google Scholar
8. J. Koekoek and R. Koekoek, Differential equations for generalized Jacobi polynomials, J. Comput. Appl. Math. 126(1) (2000) 1–31. Crossref, Web of Science, Google Scholar
9. R. Koekoek, T. H. Koornwinder, P. Lesky and R. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues (Springer-Verlag, Berlin, 2010). Crossref, Google Scholar
10. T. H. Koornwinder, Orthogonal polynomials with weight function ${(1 - x)}^{α} {(1 + x)}^{β} + M δ (x + 1) + N δ (x - 1)$ , Can. Math. Bull. 27(2) (1984) 205–214. Crossref, Web of Science, Google Scholar
11. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010). Google Scholar
12. A. Rababah, Transformation of Chebyshev–Bernstein polynomial basis, Comput. Methods Appl. Math. 3(4) (2003) 608–622. Crossref, Google Scholar
13. A. Rababah, Jacobi–Bernstein basis transformation, Comput. Methods Appl. Math. 4(2) (2004) 206–214. Crossref, Google Scholar
14. J. Rice, The Approximation of Functions: Linear Theory (Addison-Wesley, Reading, 1964). Google Scholar
15. G. Szegö, Orthogonal Polynomials, 4th edn., American Mathematical Society Colloquium Series, Vol. 23 (American Mathematical Society, Providence, 1975). Google Scholar
16. F. Yamaguchi and F. Yamaguchi, Curves and Surfaces in Computer Aided Geometric Design (Springer-Verlag, Berlin, 1988). Crossref, Google Scholar