Characterization of the generalized Chebyshev-type polynomials of first kind

Authors

  • Mohammad AlQudah Northwood University

DOI:

https://doi.org/10.14419/ijamr.v4i4.4788

Published:

2015-10-10

Keywords:

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

Abstract

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.

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