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Properties of Orthogonal Polynomials 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´eapproximants 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 Tn(x) = cos nθ where x = cos θ for n 2 N. Consider Z π cos mθ cos nθ dθ; n; m 2 N: 0 For m 6= n, Z π cos mθ cos nθ dθ 0 1 Z π = [cos(m + n)θ + cos(m − n)θ] dθ 2 0 1 sin(m + n)θ sin(m − n)θ π = + 2 m + n m − n 0 = 0: Kerstin Jordaan Properties of orthogonal polynomials The Tchebychev polynomials Tn(x) = cos nθ where x = cos θ for n 2 N. Consider Z π cos mθ cos nθ dθ; n; m 2 N: 0 For m = n, Z π Z π cos mθ cos mθ dθ = cos2 mθ dθ 0 0 1 Z π = (1 + cos 2mθ) dθ 2 0 1 sin 2mθ π = θ + 2 2m 0 π = : 2 Kerstin Jordaan Properties of orthogonal polynomials The Tchebychev polynomials Tn(x) = cos nθ where x = cos θ for n 2 N. ( Z π 0; n 6= m cos mθ cos nθ dθ = π 0 2 ; m = n: Making the substitution x = cos θ in this integral, then dx = − sin θ dθ or −dx −dx dθ = = p : sin θ 1 − x 2 Also when θ = 0, x = 1 and θ = π, x = −1 so Z π Z 1 2 −1=2 cos mθ cos nθ dθ = Tn(x)Tm(x)(1 − x ) dx 0 −1 ( 0; n 6= m = π 2 ; m = n: Kerstin Jordaan Properties of orthogonal polynomials Orthogonality Definition 1 A sequence of polynomials fpn(x)gn=0 where pn(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;::: ( Z b 0 if n 6= m pn(x) pm(x) w(x) dx = a hn 6= 0 if n = m: For Tchebychev polynomials Z 1 ( 2 −1=2 0; n 6= m Tn(x)Tm(x)(1 − x ) dx = π −1 2 ; m = n: 1 Tchebychev polynomials fTn(x)gn=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 Z b n µn := w(x)x dx; 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 2 [a; b] and w(x) > 0 for all x 2 (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 pn(x) real, it follows that Z b 2 hn = w(x)pn(x)dx 6= 0: a The sequence of polynomial is uniquely defined up to normalization. If hn = 1 for each n = 0; 1; 2;::: the sequence of polynomials is called orthonormal. If n pn = knx + lower order terms with kn = 1 for each n = 0; 1; 2;::: , the sequence is called monic. The integral Z b hPn; Pmi := Pn(x)Pm(x)w(x)dx a denotes an inner product of the polynomials Pn and Pm. 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. Z k µk = x dµ(x); k = 0; 1; 2 :::: (1) S Definition N A sequence of real polynomials fPn(x)gn=0, N 2 N [ f1g, where Pn(x) is of exact degree n, is orthogonal with repect to µ on S, if Z hPn; Pmi = Pn(x)Pm(x) dµ(x) = hnδmn; m; n = 0; 1; 2;::: N (2) S 2 where S is the support of µ and hn is the square of the weighted L -norm of Pn given by Z 2 2 hn := hPn; Pni = kPnk = (Pn(x)) dµ(x) > 0: S Kerstin Jordaan Properties of orthogonal polynomials If the measure is absolutely continuous and the distribution dµ(x) = w(x) dx, then (2) reduces to Z b pn(x) pm(x) w(x) dx = hnδmn; m; n = 0; 1; 2;::: N (3) a or equivalently (see Assignment 1, Exercise 2), Z b m x Pn(x)w(x) dx = 0; for n = 1; 2; ··· ; m < n: a If the weight function w(x) is discrete and ρi > 0 are the values of the weight at the distinct points xi , i = 0; 1; 2;:::; M; M 2 N [ f1g, then (3) takes the form M X Pn(xi )Pm(xi )ρi = hnδmn; m; n = 0; 1; 2;:::; N i=0 Kerstin Jordaan Properties of orthogonal polynomials Gram-Schmidt orthogonalisation Since the Hilbert space L2(S; µ) contains the set of polynomials, Gram-Schmidt orthogonalisation applied to the canonical basis f1; x; x 2; : : : :::g, yields a set of orthogonal polynomials on the real line. Example Take w(x) = 1 and (a; b) = (0; 1): Start with the sequence 1; x; x 2;::: : Choose p0(x) = 1: Then we have hx; p0(x)i hx; 1i 1 p1(x) = x − p0(x) = x − = x − ; hp0(x); p0(x)i h1; 1i 2 since Z 1 Z 1 1 h1; 1i = 1 dx = 1 and hx; 1i = x dx = : 0 0 2 Kerstin Jordaan Properties of orthogonal polynomials Gram-Schmidt orthogonalisation Example Further we have 2 2 2 hx ; p0(x)i hx ; p1(x)i p2(x) = x − − p1(x) hp0(x); p0(x)i hp1(x); p1(x)i hx 2; 1i hx 2; x − 1 i 1 = x 2 − − 2 x − h1; 1i 1 1 2 hx − 2 ; x − 2 i 1 1 = x 2 − − x − 3 2 1 = x 2 − x + ; 6 1 2 1 The polynomials p0(x) = 1; p1(x) = x − 2 and p2(x) = x − x + 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 3 1 p (x) = x 3 − x 2x − 3 2 20 9 2 1 p (x) = x 4 − 2x 3 + x 2 − x + 4 7 7 70 5 20 5 5 1 p (x) = x 5 − x 4 + x 3 − x 2 + x − ; 5 2 9 6 42 252 and so on. p The orthonormal polynomials would be q0(x) = p0(x)= h0 = 1; p1(x) p q1(x) = p = 2 3(x − 1=2) h1 p p2(x) 2 1 q2(x) = p = 6 5 x − x + h2 6 p p3(x) 2 3 2 3 1 p3(x) = p = 20 7 x − x + x − ; h3 2 5 20 etcetera. Kerstin Jordaan Properties of orthogonal polynomials The three-term recurrence relation The fact that hxp; qi = hp; xqi gives rise to the following fundamental property of orthogonal polynomials. Theorem A sequence of orthogonal polynomials fPn(x)g satisfies a 3-term recurrence relation of the form. Pn+1(x) = (Anx + Bn) Pn(x) − CnPn−1(x) for n = 0; 1;:::: (4) where we set P−1(x) ≡ 0 and P0(x) ≡ 1: Here, An; Bn and Cn are real constants, n = 0; 1; 2;:::. If the leading coefficient of Pn(x) is kn > 0; then kn+1 An+1 hn+1 An = ; Cn+1 = kn An hn Kerstin Jordaan Properties of orthogonal polynomials Proof Since Pn+1(x) has degree exactly (n + 1) and so does xPn(x), we can determine An such that Pn+1(x) − AnxPn(x) is a polynomial of degree at most n. Thus n X Pn+1(x) − AnxPn(x) = bk Pk (x) (5) k=0 for some constants bk . Now, if Q(x) is any polynomial of degree m < n, we know from (3) that Z b Pn(x)Q(x)w(x)dx = 0: a If we multiply both sides of (5) by w(x)Pm(x) where m 2 f0; 1 :::; n − 2g; we obtain (upon integration) Z b Z b Pn+1(x)Pm(x)w(x)dx − An xPn(x)Pm(x)w(x)dx a a n X Z b = bk Pk (x)Pm(x)w(x)dx: k=0 a Kerstin Jordaan Properties of orthogonal polynomials Proof Z b Z b Pn+1(x)Pm(x)w(x)dx − An xPn(x)Pm(x)w(x)dx (6) a a n X Z b = bk Pk (x)Pm(x)w(x)dx: k=0 a Now the left hand side of (6) is zero for each m 2 f0; 1;:::; n − 2g since then xPm(x) is a polynomial of degree (m + 1) which is less than or equal to (n − 1).
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