Characterization of the generalized Chebyshev-type polynomials of first kind

  • Authors

    • Mohammad AlQudah Northwood University
    https://doi.org/10.14419/ijamr.v4i4.4788

    Received date: May 17, 2015

    Accepted date: September 14, 2015

    Published date: October 10, 2015

  • Bernstein basis, Chebyshev polynomials, Generalized Chebyshev-type polynomials, Orthogonal polynomials.

  • Abstract

    Orthogonal polynomials have very useful properties in mathematical problems, and in recent years there has been significant development in the field of approximation theory using orthogonal polynomials. In this paper, we characterize a sequence of generalized Chebyshev-type polynomials of the first kind {Tn(M,N)(x)}n∈N∪{0}\{T_n^{(M,N)}(x)\}_{n \in \mathbb{N} \cup \{0\}}{Tn(M,N)​(x)}n∈N∪{0}​, which are orthogonal with respect to the measure

    1−x2π dx+Mδ−1+Nδ1,\frac{\sqrt{1 - x^2}}{\pi}\,dx + M\delta_{-1} + N\delta_{1},π1−x2​​dx+Mδ−1​+Nδ1​,

    where δx\delta_xδx​ is the Dirac measure and M,N≥0M, N \ge 0M,N≥0.

    We then provide a closed form of the constructed polynomials in terms of the Bernstein polynomials Bkn(x)B_k^n(x)Bkn​(x). Finally, we conclude with some results on the integration of the weighted generalized Chebyshev-type polynomials with the Bernstein polynomials.

  • References

    1. G. Farin, Curves and Surface for Computer Aided Geometric Design, 3rd ed., Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1993.
    2. R.T. Farouki, "The Bernstein polynomial basis: A centennial retrospective", Computer Aided Geometric Design, Vol. 29, No. 6, (2012), 379–419.
    3. R.T. Farouki, V.T. Rajan, "Algorithms for polynomials in Bernstein form", Computer Aided Geometric Design, Vol. 5, No. 1, (1988), 1–26.
    4. J. Hoschek, D. Lasser, Fundamentals of Computer Aided Geometric Design, A K Peters, Wellesley, MA, 1993.
    5. J. Koekoek, R. Koekoek, "Differential equations for generalized Jacobi polynomials", Journal of Computational and Applied Mathematics, Vol. 126, Issues 1 - 2, (2000), 1–31.
    6. R. Koekoek, P. Lesky, R. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer-Verlag, Berlin, 2010.
    7. T.H. Koornwinder, "Orthogonal polynomials with weight function $(1-x)^alpha(1+x)^beta + Mdelta(x+1) + Ndelta(x-1)$", Canadian Mathematical Bulletin, Vol. 27, No. 2, (1984), 205–214.
    8. F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (editors), NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010.
    9. A. Rababah, "Transformation of Chebyshev Bernstein polynomial basis", Comput. Methods Appl. Math., Vol. 3, No. 4, (2003), 608–622.
    10. J. Rice, The Approximation of Functions, Linear Theory, Vol. 1, Addison Wesley, (1964).
    11. G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications (4th Edition), Vol. 23, American Mathematical Society, Providence, RI, 1975.

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  • How to Cite

    AlQudah, M. (2015). Characterization of the generalized Chebyshev-type polynomials of first kind. International Journal of Applied Mathematical Research, 4(4), 519-524. https://doi.org/10.14419/ijamr.v4i4.4788