Generalized Chebyshev polynomials of the second kind

These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials. | Turk J Math (2015) 39: 842 – 850 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Generalized Chebyshev polynomials of the second kind Mohammad A. ALQUDAH∗ Department of Mathematics, Northwood University, Midland, MI, USA Received: • • Accepted/Published Online: Printed: Abstract: We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials. Key words: Generalized Chebyshev polynomials, Bernstein basis, Eulerian integral 1. Introduction, background and motivation Orthogonal polynomials are very important and serve to approximate other functions, where the most commonly used orthogonal polynomials are the classical orthogonal polynomials. The field of classical orthogonal polynomials developed in the late 19th century from a study of continued fractions by . Chebyshev. We have seen the significance of orthogonal polynomials, particularly in the solution of systems of linear equations and in the least-squares approximations. Meanwhile, polynomials can be represented in many different bases, such as the monomial powers, Chebyshev, Bernstein, and Hermite basis forms. Every form of polynomial basis has its advantages, and sometimes disadvantages. Many problems can be solved and many difficulties can be removed by suitable choice of basis. In this paper we characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II) (M,N ) Ur (x). These polynomials can be used to describe the approximation of continuous

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