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Orthogonal Polynomials and Their Properties

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Orthogonal Polynomials and Their Properties Republic of the Sudan Ministry of High Education and Scientific Research Nile Valley University College of Post Graduate Mathematics Department A thesis submitted partially for M.sc. in Mathematics Orthogonal Polynomials and their Properties By Rahama Abdalla Elballa Supervised by: Dr. Adam Abdella Abakar Aug 2009 Dedication To my father and mother And To my wife and brother Acknowledgement First I am thanks Allah and I am grateful to express my deep thank to my supervisor Dr/ Adam Abdella Abakar Also my thanks to all persons who helps me, and thank for all Nile Valley University staff Contents: Significance of the study Statement of the problem Hypothesis of the study Methodology of the study Chapter one: Basic Concepts 1-1 Introduction 1-2 Objectives Chapter two: Orthogonal Polynomials Sequences 2-1 Definition 2-2 Standardization 2-3 General properties of orthogonal polynomials sequences 2- 3 -1 Recurrence relations 2-3-2 Existence of roots 2-3-3 Interlacing of roots 2-4 Differential equations leading to orthogonal polynomials 2-4-1 Rodrigue's formula 2-4- 2 The number 2-4-3 Second form for the differential equation 2-4-4 Third form for the differential equation 2-4-5 Orthogonality Chapter three :The Classical Orthogonal Polynomials 3-1 Jacobi polynomials 3-2 Gegenbauer polynomials 3-3 Legendre polynomials 3-3-1 Associated Legendre polynomial 3-4 Chebyshev polynomials 3-5 Laguerre polynomials 3-6 Hermite polynomials Chapter four: Constructing orthogonal polynonlials ' using moment 4-1 Significance of the moment 4-2 Constructing orthogonal polynomials 4-3 Table of classical orthogonal polynomials. Chapter five: Some application 5-1 Mathematical applications 5-1-1 Numerical analysis 5-1-2 Theory of random matrices 5-1-3 Computer approximations 5-2 Physics and Engineering applications 5-2-1 Electromagnetism 5-2-3 Classical mechanics 5-2-3 Communition Conclusion & Recommendation References Abstract: Orthogonal polynomials are classes of polynomials {Pn(x)} defined over an interval that obey the property of orthogonality. The main contributions of this work concern the description of a small sample of what is known about specific sets of orthogonal polynomials the general basic concepts, and the constructing of orthogonal polynomials. by using the moment, and also the presentation of some uses or applications in the field of Mathematics, Physics, and Engineering. الخﻻصة كثيرات الحدود المتعامدة هي أنواع من كثيرات الحدود يمكن تعريفها عمى أنها الفترة التي تتفق مع خواص التعامد. إن اﻹسهامات الرئيسية في هذا العمل تتضمن وصف نموذج صغير فيما يعرف بمجموعات محددة من كثيرات الحدود المتعامدة، المفاهيم اﻷساسية العامة، وتكوين كثي ارت الحدود المتعامدة باستخدام العزوم وأيضاً تقديم بعض اﻻستخدامات أو التطبيقات في مجال الرياضيات والفيزياء والهندسة. Chapter One Basic Concepts 1-Introduction: Orthogonal polynomials seelned to have first been developed by Legendre and Laplace in their work of celestial mechanics. One of the developments was work of Laplace on probability theory. Both of these dealt with specific sets of orthogonal polynomials, first polynomials orthogonal with respect to a symmetric beta distribution on (-1,1) and then the normal distribution on the whole real line. The general theory starts with Chebyshev in the 1850s. Orthogonal polynomials are classes of polynomials {Pn(x)} defined over an interval that obey the property of orthogonality. In the present chapter we shall introduce the corresponding concepts and notations. Two polynomials Pm(x) and Pn(x) which are real-valued continuous functions defined on the interval a < x < b and are such that the integral of the product Pm(x) Pn(x) over that interval exists. We shall denote this integral by ( P m,P n) such that: (Pm, pn) = ( 1-1) where ( Pm, Pn) means the inner product space of continuous functions Pm, Pn with the following properties: 1- For any scalars and and any polynomials Pm(x), Pn(x),Ps(x) in polynomial space V, then ( Pm (x) + Pn (x),Ps (x)) = (Pm (x ),Ps (x ))+ (Pn (x ),Ps(x) (linearity) 2- For any polynomials Pm(x) and Pn(x) in V, (Pm, Pn) = (Pn, Pm) (symmetry) 3- For any Pm(x) in V, ( Pm, Pn) > 0 and (positive-defmiteness) (Pm, Pn) = 0 if and only if Pm =0 The functions are said to be orthogonal on the interval a <x <b . if: (Pm,Pn)= ,(m n) (1-2) [1][4] 1 A set of real-valued functions P1(x), P2(x), ............... is called an orthogonal set of functions on an interval a <x <b if these functions are defined on that interval and if all the integral(Pm, Pn) exist and are zero for all pairs of distinct functions in the set. The non negative square root of (Pm ,P n) is called the norm Pm(x) and is generally denoted by IlPmll , where, (1-3) It is clear that an orthogonal set P I,P 2, ... on an interval a x b whose functions have norm 1 satisfies the relations: (Pm,Pn) = = (1-4) Such a set is called an orthonormal set of functions on the interval a < x <b . Obviously , from an orthogonal set we may obtain ·-an orthonormal set by dividing each function by its norm on the interval under consideration. for example, the function Pm(x)= sinmx, m=1,2, .... Form an orthogonal set on the interval -Il < x <Il, because (Pm, Pn) = = (1-5) The norm ||Pm|| equals , because 2 (Pm, Pn) = ||Pm|| = , (m=1,2,…..) So the corresponding orthonormal set consists of the functions: Then the orthogonal sets yield important types of series development. In fact ,let P1(x), P2(x), .... be any orthogonal set of functions on an interval a < x <b and let f(x) be a given function which can be represented in term of the Pj's by a convergence series: (1-6)[1][3] 2 This series is called the generalized Fourier series of f(x) and its coefficients cl,c2, .. : ... are called the Fourier constants of f(x) with respect to that orthogonal set of functions. In fact, multiplying both sides of (1-6) by Pm(x) (m fixed), integrating over a <x <b, and assuming that term-by-term integration is permissible, we have :- (f,Pm)= = Integral for which n=m equals (Pm ,Pm) = || Pm||2,wehere as all the other integrals on the right are zero, because of the 2 othogonality. Hence (f, Pm) = cm|| Pm|| , the desired formula for the Fourier constant is: (1-7) Some important sets of real functions Pl,P2, ..... occurring in applications are not orthogonal but have the property that for some function W(x), (1-8) Such a set is then said to be orthogonal with respect to the weight function W(x) on the interval a <x <b. The norm of Pm is now defined as : (1-9) And if the norm of each function Pm is 1 , the set is said to be orthonormal on that interval with respect to W(x).If we sethm= Pm then (1-8) becomes: (1-10) That is the functions hm form an orthogonal set in the usual sense. Various important orthogonal sets of functions arise as solutions of second-order differential equations of the form in some interval a <x <b, satisfying conditions of the form : (a) K1y+K2Y'=0 at x=a (b) L1y+L2y'=0 at x=b; (1-12) [2][3][5] 3 Where is a parameter , and K1, K2,L1, L2 are given are given real constants, at least one in each conditions (1-12) being different from zero. The equation (1-11) is known as the Sturm-Liouville equation, and we shall see that Legendre's equation, and other important equations can be written in the form (1-11). The conditions (1-12)refer to the boundary points x=a and x=b of that interval and, therefore, are called boundary conditions. A differential equation(1-11) with respect to the boundary conditions (1-12) is called Liouvilleproblem. This problem has the trivial solution y=0 for any value of the parameter . Solution y 0 are called characteristic functions or eigenfunctions of the problem, and thy values of for which such solutions exist. For example: if we need to find the eigenvalues and eigenfunctions of the Sturm-Liouville problem: (a) y"+ y=0 (b) y(o) = 0, y( ) =0 (1-13) For negative the general solution of the equation is: vx -vx y(x) = c1e + c2e From (l-13b) we obtain cl=c2=0 and q=0, which is not an eigenfunction. For = 0 the situation is similar. For positive = v2_ the general solution is y(x) =Acosvx+B sin vx. From the first boundary condition we obtain y(0)=A=0. The second boundary condition then yields: y( ) = B sin v or v = 0,±1,±2, for v=O we have y=0. For =v2 =1,4,9,16, .. , taking B=I,we obtain y(x) = sin vx v = 1,2, . These are the eigenfunctions of the problem, and the eigenvalues are = v2 where v = 1,2, .. It can be shown that under rather general conditions on the functions p, q , and r in (1-11) the Sturm-Liouville problem (1-11), (1-12) has infinitely many eigenvalues. Furthermore, theeigenfunctions are orthogonal, as follows: Theorem 1: (orthogonality).Let the functions p, q and r in the strum-Liouville equation (1-11) be real-value and continuous 4 on he interval a <x <b . Let Ym(x) and Yn(x) be eigenfunctions of the Sturm-Liouville problem (1-11),(1-12) corresponding to distinct eigenvalues , , respectively. Then Ym and Yn are orthogonal on that interval with respect to the weight function W.[3][4][5] If r(a) =0,.
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