Characterization of the generalized Chebyshev-type polynomials of first kind


  • Mohammad AlQudah Northwood University





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


Orthogonal polynomials have very useful properties in the mathematical problems, so recent years have seen a great deal in the  field of approximation theory using orthogonal polynomials. In this paper, we characterize a sequence of the generalized Chebyshev-type polynomials of the first kind  \(\left\{\mathscr{T}_{n}^{(M,N)}(x)\right\}_{n\in\mathbb{N}\cup\{0\}},\)  which are orthogonal with respect to the measure \(\frac{\sqrt{1-x^{2}}}{\pi}dx+M\delta_{-1}+N\delta_{1},\) where \(\delta_{x}\) is a singular Dirac measure and \(M,N\geq 0.\) Then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials \(B_{k}^{n}(x).\)

We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.


[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, em 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 ", 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] 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. Szeg, Orthogonal Polynomials, American Mathematical Society Colloquium Publications (4th Edition), Vol. 23, American Mathematical Society, Providence, RI, 1975.

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