Properties of orthogonal polynomials Kerstin Jordaan University of - PowerPoint PPT Presentation

Properties of orthogonal polynomials Kerstin Jordaan University of South Africa LMS Research School University of Kent, Canterbury Kerstin Jordaan Properties of orthogonal polynomials

Outline 1 Orthogonal polynomials Gram-Schmidt orthogonalisation The three-term recurrence relation Jacobi operator Hankel determinants Hermite and Laguerre polynomials 2 Properties of classical orthogonal polynomials 3 Quasi-orthogonality and semiclassical orthogonal polynomials 4 The hypergeometric function 5 Convergence of Pad´ e approximants for a hypergeometric function Kerstin Jordaan Properties of orthogonal polynomials

The pioneer of orthogonality Chebyshev Chebychev Chebyshov Tchebychev Tchebycheff Tschebyscheff Murphy [1835] first defined orthogonal functions, Tchebychev realised their importance. His work since 1855 was motivated by the analogy with Fourier Series and by the theory of continued fractions and approximation theory. Kerstin Jordaan Properties of orthogonal polynomials

The Tchebychev polynomials T n ( x ) = cos n θ where x = cos θ for n ∈ N . Consider � π cos m θ cos n θ d θ, n , m ∈ N . 0 For m � = n , � π cos m θ cos n θ d θ 0 � π = 1 [cos( m + n ) θ + cos( m − n ) θ ] d θ 2 0 � sin( m + n ) θ � π = 1 + sin( m − n ) θ 2 m + n m − n 0 = 0 . Kerstin Jordaan Properties of orthogonal polynomials

The Tchebychev polynomials T n ( x ) = cos n θ where x = cos θ for n ∈ N . Consider � π cos m θ cos n θ d θ, n , m ∈ N . 0 For m = n , � π � π cos 2 m θ d θ cos m θ cos m θ d θ = 0 0 � π = 1 (1 + cos 2 m θ ) d θ 2 0 � � π = 1 θ + sin 2 m θ 2 2 m 0 = π 2 . Kerstin Jordaan Properties of orthogonal polynomials

The Tchebychev polynomials T n ( x ) = cos n θ where x = cos θ for n ∈ N . � � π 0 , n � = m cos m θ cos n θ d θ = π 2 , m = n . 0 Making the substitution x = cos θ in this integral, then dx = − sin θ d θ or d θ = − dx − dx sin θ = √ 1 − x 2 . Also when θ = 0, x = 1 and θ = π , x = − 1 so � π � 1 T n ( x ) T m ( x )(1 − x 2 ) − 1 / 2 dx cos m θ cos n θ d θ = − 1 0 � 0 , n � = m = π 2 , m = n . Kerstin Jordaan Properties of orthogonal polynomials

Orthogonality Definition A sequence of polynomials { p n ( x ) } ∞ n =0 where p n ( x ) is of exact degree n , is called orthogonal on the interval ( a , b ) with respect to the positive weight function w ( x ) if, for m , n = 0 , 1 , 2 , . . . � � b 0 if n � = m p n ( x ) p m ( x ) w ( x ) dx = h n � = 0 if n = m . a For Tchebychev polynomials � � 1 0 , n � = m T n ( x ) T m ( x )(1 − x 2 ) − 1 / 2 dx = π 2 , m = n . − 1 Tchebychev polynomials { T n ( x ) } ∞ n =0 are orthogonal on the interval [ − 1 , 1] with respect to the positive weight function (1 − x 2 ) − 1 / 2 . Kerstin Jordaan Properties of orthogonal polynomials

The interval ( a , b ) is called the interval of orthogonality and need not be finite. With due attention to convergence, either or both endpoints of the interval of orthogonality may be taken to be infinite. The limits of integration are important but the form in which the interval of orthogonality is stated is not vital. The weight function w(x) should be continuous and positive on ( a , b ) so that the moments � b w ( x ) x n dx , µ n := n = 0 , 1 , 2 . . . a exist. The weight function w ( x ) does not change sign on the interval of orthogonality by assumption may vanish at the finite endpoints (if any) of the interval of orthogonality w ( x ) ≥ 0 for all x ∈ [ a , b ] and w ( x ) > 0 for all x ∈ ( a , b ) is the usual definition of a weight function Kerstin Jordaan Properties of orthogonal polynomials

More remarks Because we have taken w ( x ) > 0 on ( a , b ) and p n ( x ) real, it follows that � b w ( x ) p 2 h n = n ( x ) dx � = 0 . a The sequence of polynomial is uniquely defined up to normalization. If h n = 1 for each n = 0 , 1 , 2 , . . . the sequence of polynomials is called orthonormal. If p n = k n x n + lower order terms with k n = 1 for each n = 0 , 1 , 2 , . . . , the sequence is called monic. The integral � b � P n , P m � := P n ( x ) P m ( x ) w ( x ) dx a denotes an inner product of the polynomials P n and P m . Kerstin Jordaan Properties of orthogonal polynomials

More generally Let µ be a positive Borel measure with support S defined on R for which moments of all orders exist, i.e. � x k d µ ( x ) , µ k = k = 0 , 1 , 2 . . . . (1) S Definition A sequence of real polynomials { P n ( x ) } N n =0 , N ∈ N ∪ {∞} , where P n ( x ) is of exact degree n , is orthogonal with repect to µ on S , if � � P n , P m � = P n ( x ) P m ( x ) d µ ( x ) = h n δ mn , m , n = 0 , 1 , 2 , . . . N (2) S where S is the support of µ and h n is the square of the weighted L 2 -norm of P n given by � h n := � P n , P n � = � P n � 2 = ( P n ( x )) 2 d µ ( x ) > 0 . S Kerstin Jordaan Properties of orthogonal polynomials

If the measure is absolutely continuous and the distribution d µ ( x ) = w ( x ) dx , then (2) reduces to � b p n ( x ) p m ( x ) w ( x ) dx = h n δ mn , m , n = 0 , 1 , 2 , . . . N (3) a or equivalently (see Assignment 1, Exercise 2), � b x m P n ( x ) w ( x ) dx = 0 , for n = 1 , 2 , · · · ; m < n . a If the weight function w ( x ) is discrete and ρ i > 0 are the values of the weight at the distinct points x i , i = 0 , 1 , 2 , . . . , M , M ∈ N ∪ {∞} , then (3) takes the form � M P n ( x i ) P m ( x i ) ρ i = h n δ mn , m , n = 0 , 1 , 2 , . . . , N i =0 Kerstin Jordaan Properties of orthogonal polynomials

Gram-Schmidt orthogonalisation Since the Hilbert space L 2 ( S , µ ) contains the set of polynomials, Gram-Schmidt orthogonalisation applied to the canonical basis { 1 , x , x 2 , . . . ... } , yields a set of orthogonal polynomials on the real line. Example Take w ( x ) = 1 and ( a , b ) = (0 , 1) . � � 1 , x , x 2 , . . . Start with the sequence . Choose p 0 ( x ) = 1 . Then we have � p 0 ( x ) , p 0 ( x ) � p 0 ( x ) = x − � x , 1 � � x , p 0 ( x ) � � 1 , 1 � = x − 1 p 1 ( x ) = x − 2 , since � 1 � 1 x dx = 1 � 1 , 1 � = 1 dx = 1 and � x , 1 � = 2 . 0 0 Kerstin Jordaan Properties of orthogonal polynomials

Gram-Schmidt orthogonalisation Example Further we have � x 2 , p 0 ( x ) � � x 2 , p 1 ( x ) � p 2 ( x ) = x 2 − � p 0 ( x ) , p 0 ( x ) � − � p 1 ( x ) , p 1 ( x ) � p 1 ( x ) � � � x 2 , x − 1 = x 2 − � x 2 , 1 � 2 � x − 1 � 1 , 1 � − � x − 1 2 , x − 1 2 2 � � � = x 2 − 1 x − 1 3 − 2 = x 2 − x + 1 6 , 2 and p 2 ( x ) = x 2 − x + 1 The polynomials p 0 ( x ) = 1 , p 1 ( x ) = x − 1 6 are the first three monic orthogonal polynomials on the interval (0, 1) with respect to the weight function w ( x ) = 1 . Kerstin Jordaan Properties of orthogonal polynomials

Example Repeating this process we obtain p 3 ( x ) = x 3 − 3 2 x 2 x − 1 20 p 4 ( x ) = x 4 − 2 x 3 + 9 7 x 2 − 2 7 x + 1 70 p 5 ( x ) = x 5 − 5 2 x 4 + 20 9 x 3 − 5 6 x 2 + 5 1 42 x − 252 , and so on. The orthonormal polynomials would be q 0 ( x ) = p 0 ( x ) / √ h 0 = 1 , √ q 1 ( x ) = p 1 ( x ) √ h 1 = 2 3( x − 1 / 2) � � √ q 2 ( x ) = p 2 ( x ) x 2 − x + 1 √ h 2 = 6 5 6 � � √ p 3 ( x ) = p 3 ( x ) x 2 − 3 2 x 2 + 3 5 x − 1 √ h 3 = 20 7 , 20 etcetera. Kerstin Jordaan Properties of orthogonal polynomials

The three-term recurrence relation The fact that � xp , q � = � p , xq � gives rise to the following fundamental property of orthogonal polynomials. Theorem A sequence of orthogonal polynomials { P n ( x ) } satisfies a 3-term recurrence relation of the form. P n +1 ( x ) = ( A n x + B n ) P n ( x ) − C n P n − 1 ( x ) for n = 0 , 1 , . . . . (4) where we set P − 1 ( x ) ≡ 0 and P 0 ( x ) ≡ 1 . Here, A n , B n and C n are real constants, n = 0 , 1 , 2 , . . . . If the leading coefficient of P n ( x ) is k n > 0 , then A n = k n +1 k n , C n +1 = A n +1 h n +1 A n h n Kerstin Jordaan Properties of orthogonal polynomials

Proof Since P n +1 ( x ) has degree exactly ( n + 1) and so does xP n ( x ), we can determine A n such that P n +1 ( x ) − A n xP n ( x ) is a polynomial of degree at most n . Thus n � P n +1 ( x ) − A n xP n ( x ) = b k P k ( x ) (5) k =0 for some constants b k . Now, if Q ( x ) is any polynomial of degree m < n , we know from (3) that � b P n ( x ) Q ( x ) w ( x ) dx = 0 . a If we multiply both sides of (5) by w ( x ) P m ( x ) where m ∈ { 0 , 1 . . . , n − 2 } , we obtain (upon integration) � b � b P n +1 ( x ) P m ( x ) w ( x ) dx − A n xP n ( x ) P m ( x ) w ( x ) dx a a � b � n = b k P k ( x ) P m ( x ) w ( x ) dx . a k =0 Kerstin Jordaan Properties of orthogonal polynomials