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Orthogonal Polynomials and Classical Orthogonal Polynomials International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 10, October 2018, pp. 1613–1630, Article ID: IJMET_09_10_164 Available online at http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed ORTHOGONAL POLYNOMIALS AND CLASSICAL ORTHOGONAL POLYNOMIALS DUNIA ALAWAI JARWAN Education for Girls College, Al-Anbar University, Ministry of Higher Education and Scientific Research, Iraq ABSTRACT The focus of this project is to clarify the concept of orthogonal polynomials in the case of continuous internal and discrete points on R and the Gram – Schmidt orthogonalization process of conversion to many orthogonal limits and the characteristics of this method. We have highlighted the classical orthogonal polynomials as an example of orthogonal polynomials because of they are great importance in physical practical applications. In this project, we present 3 types (Hermite – Laguerre – Jacobi) of classical orthogonal polynomials by clarifying the different formulas of each type and how to reach some formulas, especially the form of the orthogonality relation of each. Keywords: Polynomials, Classical Orthogonal, Monic Polynomial, Gram – Schmidt Cite this Article Dunia Alawai Jarwan, Orthogonal Polynomials and Classical Orthogonal Polynomials, International Journal of Mechanical Engineering and Technology, 9(10), 2018, pp. 1613–1630. http://iaeme.com/Home/issue/IJMET?Volume=9&Issue=10 1. INTRODUCTION The mathematics is the branch where the lots of concepts are included. An orthogonality is the one of the concept among them. Here we focuse on the orthogonal polynomial sequence. The orthogonal polynomial are divided in two classes i.e. classical orthogonal polynomials, Discrete orthogonal polynomials and Sieved orthogonal polynomials .There are different types of classical orthogonal polynomials such that Jacobi polynomials, Associated Laguerre polynomials and Hermite polynomials. The Jacobi polynomials are subdivided as Gegerbauer, legendre, and Chebyshev (it has its own two types). The plain Laguerre polynomial is the subclass of the Laguerre polynomials. Discrete orthogonal polynomials are based on the discrete values. In some cases these values are finite rather than an infinite sequence. The Racah polynomial is the example of discrete orthogonal polynomials. It has some special case like Hahn polynomials and dual Hahn polynomials with Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. The Sieved orthogonal polynomial is the one of orthogonal polynomial. It is classified as are sieved ultra spherical polynomials, sieved Jacobi polynomials, and polynomials. It includes modified recurrence relations. http://iaeme.com/Home/journal/IJMET 1613 [email protected] Dunia Alawai Jarwan 1.1. Definition of polynomial For a general polynomial of degree n, n n −1 Pn (x ) = a n X + a n −1 X + ..... + a 0 , an ≠ 0 [Bickel et al. 2000] We call an the leading coefficient of the polynomial. 1.2. Definition (Monic polynomial) A polynomial is monic if the coefficient of its leading term is 1. % Pn ( x ) n an − n −1 ao P ( x ) = = X + 1 X + .....+ n a a a We denote by n n n [Bickel et al. 2000], 1.3. Definition (Inner product) Let V is a real vector space. An Inner product is a real – valued function .,. on V ×V such that, for all f , g , h ∈V , I. f + g , h = f , h + g,h λ f ,g = λ f ,g II. for λ ∈ III. f , g = g, f f f IV. , g ≥ 0 and , f = 0 iff f =0 see [Milovanovic and Gradimir, 2013; Axler and Sheldon, 2015] Let us take V is our vector space, the X = (x 1 , x 2 ,...,x n ) and Y = ( y1 , y 2 ,..., y n ). Then the following is an inner product: n x , y = ∑x i y i [Axler and Sheldon, 2015] i =1 Remark 2 2 The length of vectors X by X = x , x = x 1 + ..... + x n and If X = 1 , we call X a unit vector and X is said be normalized. See [Axler and Sheldon, 2015] * The set of real – valued functions that defined on the continuous interval [a ,b ], denoted b C [a ,b ] , is a vector space. And f , g = ∫f ( x ) g ( x )dx , is an inner product. See [Milovanovic a and Gradimir, 2013] 1.4. Definition (Orthogonality in general linear spaces) Two elements a ,b of a vector space are orthogonal with respect to a given inner product .,. if a, b = 0 Definition: Orthonormal An orthonormal set is a subsect S of an inner product space, such that for all X ,y ∈S http://iaeme.com/Home/journal/IJMET 1614 [email protected] Orthogonal Polynomials and Classical Orthogonal Polynomials 0 if x ≠ y x, y = if X y 1 = [Axler and Sheldon, 2015] Remark The set of the (standard basis) vectors {e1 = (1,0,0 ), e 2 = (0,1,0 ),....,en } forms an orthonormal basis of Rn. Let υ1 , υ 2 ,...,υ k be mutually orthogonal vectors, then 2 2 2 υ + υ + ... + υ = υ + ....+ υ 1 2 k 1 k see [Axler and Sheldon, 2015] 2. ORTHOGONAL POLYNOMIAL 2.1. Orthogonality polynomial on Intervals A set of orthogonal polynomials is an (infinite of finite) sequence of polynomials, P0 (x ) , P1 (x ), P2 ( x ) ,...., where Pn (x ) has degree n and any two polynomials in the set are orthogonal to each other on the C [a ,b ] , that is b P i , P j = ∫ P i ( x ) P j ( x ) a dx =0 for i ≠ j see [Agarwal and Gradimir, 1991; Diekema and Tom, 2012; Jia and Yan-Bin, 2016] Otherwise, ∞ A sequence of polynomials P x with degree P (x ) = n for each n is called { n ( )}n= 0 n orthogonal with respect to the weight function W (x ) on the C [a, b ] if the inner product of the polynomials Pi , Pj P P P x P x w x dx h h n , m = ∫ n ( ) m ( ) ( ) = n δn ,m , n ≥ 0 0 , n ≠ m δnm = 1 , n = m With where δn ,m is the Kronecker delta function The weight function w (x ) should be continuous and positive on C [a ,b ] such that the b n moments µn = ∫w ( x )X dx , n = 0,1, 2,3,... exist. a Remark: Note that the range of the integral may be infinite. See [Bickel et al, 2000; Everitta et al, 2001; Antonia et al, 2009; Stange, 2010; Diekema and Tom, 2012; Duenas et al, 2013; Sadjang et al, 2015] 2.2. Orthogonality polynomial on Finite point sets Let X be a finite set of discrete points on R, or a countable infinite set of discrete points on R and w x , x ∈X be a set of positive constant then a system of polynomials {Pn (x )}, n = 0,1,2,.... is said to be orthogonal on X with respect the weights w x if http://iaeme.com/Home/journal/IJMET 1615 [email protected] Dunia Alawai Jarwan P n ,P m = ∑ P n (x )P m (x w) x = 0 , n ≠ m x ∈X when X is infinite ∑ Pn (x ) Pm ( x W) x = 0 , n , m = 0,1, 2,.... N , n ≠ m , or x ∈X When X is a finite set of N +1 discrete points. n In the former case we also require ∑ X w x < ∞ see ([Area et al, 1998; x ∈X Everitta et al., 2001; Arvesu et al, 2003; Clarkson, 2013; Diekema and Tom 2012; Duenas et al., 2013; Eisinberg and Giuseppe, 2007]) 2.3. Properties of orthogonal polynomials (1) All sets of orthogonal polynomials have a number of fascinating properties: Any polynomial f ( x ) of degree n can be expanded in terms of P0 , P1 ,....,Pn , that is, there exist coefficients ai such that n f ( x ) = ∑ai Pi ( x ) i =0 see [Area et al, 1998] (2) Given an orthogonal set of polynomials {P0 (x ) , P1 (x ),....} each polynomial, Pk (x ) is orthogonal to any polynomial of degree < k . (3) Any orthogonal set of polynomials {P0 ( x ), P1 ( x ) ,....} has a recurrence formula that relates any three consecutive polynomials in the sequence, that is, the relation P = a X + b P −C P n +1 ( n n ) n n n −1 exits, Where a , b ,c coefficients depend on n . Such a recurrence formula is often used to generate higher order members in the set. (4) Each polynomial in {P0 ( x ), P1 (x ),....} has all n of its roots real, distinct and strictly with in the internal of orthogonality (i.e. not on its ends). th th (5) Furthermore the roots of the n degree polynomial, Pn lie strictly inside the roots of the ( n +1) degree polynomial Pn+1 . (see [Area et al, 1998; Antonia, 2009; Masjed-jamei, 2006]) 2.4. Gram - Schmidt orthogonalization process Any infinite sequence of polynomials {Pn }with Pn having degree n forms a basis for the infinite dimensional vector space of all polynomials. Such a sequence can be turned into an orthogonal basis using the Gram – Schmidt orthogonalization process, by projecting out the components of each polynomial that are orthogonal to the polynomials already chosen. Let assume V as a vector space along with an inner product. Let B = {X 1, X 2,...., X n } as a basis for V , and not necessarily orthogonal. An orthogonal basis \ B = {ν1 ,ν 2 , ν3 ,....,ν n } may be constructed from B as follows: Then V 1 = X 2 http://iaeme.com/Home/journal/IJMET 1616 [email protected] Orthogonal Polynomials and Classical Orthogonal Polynomials x 2 ,ν 1 V = X − proj X = X − ν 2 2 ν1 2 2 1 ν 1, ν1 x 3 ,ν 2 x3 ,ν1 V = X − projν X − projν X = X − ν − ν 3 3 2 3 1 3 3 ν ,ν 2 ν ,ν 1 2 2 1 1 .
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