Australian Journal of Basic and Applied Sciences, 5(9): 2083-2095, 2011 ISSN 1991-8178

1. Australian Journal of Basic and Applied Sciences, 5(9): 2083-2095, 2011 ISSN 1991-8178 A Survey On Discrete Inner Products Of Chebyshev Polynomials A. Arzhang Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. Abstract: We introduce several kinds of inner products in the set of the zeros of Chebyshev polynomials of four kinds. We give a survey on discrete orthogonality relations that exist between the four kinds of Chebyshev polynomials in the set of the zeros of these polynomials. In each theorem, we establish relations step by step that are used for proving the next relations. Finally, we give an application of the obtained formulas. Key words: Chebyshev polynomials; Discrete orthogonality; Fredholm integral equations; Bubnov-Galerkin method; Trigonometrical summations. MSC: 33D45; 33D52; 45B05. INTRODUCTION Chebyshev polynomials are well-known special orthogonal polynomials. These polynomials have a great impotence in the approximation theory. These polynomials are widely used in many areas of numerical analysis such as, uniform approximation, numerical integration, least-squares approximation, numerical solution of ordinary and partial differential equations, numerical solution of integral equations, and so on (Delves, and Mohammed, 1985: Rivlin 1974) Numerous articles and books have been written about this topics. Many certain basic fascinating properties of Chebyshev polynomials are investigated (Arfken, and Weber, 2001: Gradshteyn, and Ryzhik, 2007: Mason, and Handscomb 2003: Snyder 1966). Chebyshev polynomials of the first and second kind have been presented by many authors, but the third and forth kind are used less than the first and second kind. In practice, it is neither convenient nor efficient to work out with the first kind. So, in this paper, we discuss discrete orthogonality of the four kinds in the set of the zeros of four kinds. Chebyshev polynomials of the first kind are also known as the optimal approximation polynomials on the interval (Abramowitz, and Stegun 1972: Abramowitz, and Stegun 1972) and interpolation in the zeros of These polynomials are an interesting subject (Mason, and Handscomb 2003: Rivlin, 1990: Rivlin, 1974). U ( x ) = sin ( n + 1) q , n sin q cos ( n + 1 )q Chebyshev polynomials of the first, second, third and fourth kinds of degree n , are defined by T n ( x ) = cos n q , V ( x ) = 2 , cos q 2 sin ( n + 1 )q n W ( x ) = 2 , sin q 2 (1) respectively, where x = cosq and 0 £q£p. The four kinds of Chebyshev polynomials functions Tn ,U n ,Vn ,Wn are orthogonal with respect to the weight n ò-1 ïp m = n = 0, m = n ¹ 0, 2 1 1 -x 2 dx = íp 1 +x on the interval [-1,1] , i.e. 1 T m ( x ) T n ( x ) ì ï 0 m ¹n , ï (2) ò- 1 1 -x 2 U ( x)U n ( x)dx = pd , 2 mn 1 -x 2 , , 1 +x , 1 , 1 -x 2 m, n = 0,1,L, (3) Corresponding Author: Arzhang Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. E-mail: Address: arzhang@kiau.ac.ir 2083 1 -x 1 -x , m