Orthogonal polynomials

Transcription

1 Orthogonal polynomials Gérard MEURANT October, 2008

2 1 Definition 2 Moments 3 Existence 4 Three-term recurrences 5 Jacobi matrices 6 Christoffel-Darboux relation 7 Examples of orthogonal polynomials 8 Variable-signed weight functions 9 Matrix orthogonal polynomials

3 Definition [a, b] = finite or infinite interval of the real line Definition A Riemann Stieltjes integral of a real valued function f of a real variable with respect to a real function α is denoted by b a f (λ) dα(λ) (1) and is defined to be the limit (if it exists), as the mesh size of the partition π of the interval [a, b] goes to zero, of the sums where c i [λ i, λ i+1 ] {λ i } π f (c i )(α(λ i+1 ) α(λ i ))

4 if f is continuous and α is of bounded variation on [a, b] then the integral exists α is of bounded variation if it is the difference of two nondecreasing functions The integral exists if f is continuous and α is nondecreasing In many cases Riemann Stieltjes integrals are directly written as b a f (λ) w(λ)dλ where w is called the weight function

5 Moments and inner product Let α be a nondecreasing function on the interval (a, b) having finite limits at ± if a = and/or b = + Definition The numbers µ i = b a λ i dα(λ), i = 0, 1,... (2) are called the moments related to the measure α Definition Let P be the space of real polynomials, we define an inner product (related to the measure α) of two polynomials p and q P as p, q = b a p(λ)q(λ) dα(λ) (3)

6 The norm of p is defined as ( b p = a ) 1 p(λ) 2 2 dα(λ) (4) We will consider also discrete inner products as p, q = m j=1 p(t j )q(t j )w 2 j (5) The values t j are referred as points or nodes and the values wj 2 the weights are

7 We will use the fact that the sum in equation (5) can be seen as an approximation of the integral (3) Conversely, it can be written as a Riemann Stieltjes integral for a measure α which is piecewise constant and has jumps at the nodes t j (that we assume to be distinct for simplicity), see Atkinson; Dahlquist, Eisenstat and Golub; Dahlquist, Golub and Nash 0 if λ < t 1 α(λ) = i j=1 [w j] 2 if t i λ < t i+1 i = 1,..., m 1 m j=1 [w j] 2 if t m λ

8 There are different ways to normalize polynomials: A polynomial p of exact degree k is said to be monic if the coefficient of the monomial of highest degree is 1, that is p(λ) = λ k + c k 1 λ k Definition The polynomials p and q are said to be orthogonal with respect to inner products (3) or (5), if p, q = 0 The polynomials p in a set of polynomials are orthonormal if they are mutually orthogonal and if p, p = 1 Polynomials in a set are said to be monic orthogonal polynomials if they are orthogonal, monic and their norms are strictly positive

9 The inner product, is said to be positive definite if p > 0 for all nonzero p P A necessary and sufficient condition for having a positive definite inner product is that the determinants of the Hankel moment matrices are positive µ 0 µ 1 µ k 1 µ 1 µ 2 µ k det > 0, k = 1, 2, µ k 1 µ k µ 2k 2 where µ i are the moments of definition (2)

10 Existence of orthogonal polynomials Theorem If the inner product, is positive definite on P, there exists a unique infinite sequence of monic orthogonal polynomials related to the measure α See Gautschi

11 Minimization properties Theorem If q k is a monic polynomial of degree k, then b min q q k 2 (λ) dα(λ), k a is attained if and only if q k is a constant times the orthogonal polynomial p k related to α See Szegö We have defined orthogonality relative to an inner product given by a Riemann Stieltjes integral but, more generally, orthogonal polynomials can be defined relative to a linear functional L such that L(λ k ) = µ k Two polynomials p and q are said to be orthogonal if L(pq) = 0 One obtains the same kind of existence result, see the book by Brezinski

12 Three-term recurrences The main ingredient is the following property for the inner product λp, q = p, λq Theorem For monic orthogonal polynomials, there exist sequences of coefficients α k, k = 1, 2,... and γ k, k = 1, 2,... such that p k+1 (λ) = (λ α k+1 )p k (λ) γ k p k 1 (λ), k = 0, 1,... (6) p 1 (λ) 0, p 0 (λ) 1. where α k+1 = λp k, p k, k = 0, 1,... p k, p k γ k = p k, p k, k = 1, 2,... p k 1, p k 1

13 Proof. A set of monic orthogonal polynomials p j is linearly independent Any polynomial p of degree k can be written as p = k ω j p j, j=0 for some real numbers ω j p k+1 λp k is of degree k k 2 p k+1 λp k = α k+1 p k γ k p k 1 + δ j p j (7) Taking the inner product of equation (7) with p k λp k, p k = α k+1 p k, p k j=0

14 Multiplying equation (7) by p k 1 λp k, p k 1 = γ k p k 1, p k 1 But, using equation (7) for the degree k 1 λp k, p k 1 = p k, λp k 1 = p k, p k we multiply equation (7) with p j, j < k 1 λp k, p j = δ j p j, p j The left hand side of the last equation vanishes For this, the property λp k, p j = p k, λp j is crucial Since λp j is of degree < k, the left hand side is 0 and it implies δ j = 0, j = 0,..., k 2

15 There is a converse to this theorem It is is attributed to J. Favard whose paper was published in 1935, although this result had also been obtained by J. Shohat at about the same time and it was known earlier to Stieltjes Theorem If a sequence of monic orthogonal polynomials p k, k = 0, 1,... satisfies a three term recurrence relation such as equation (6) with real coefficients and γ k > 0, then there exists a positive measure α such that the sequence p k is orthogonal with respect to an inner product defined by a Riemann Stieltjes integral for the measure α

16 Orthonormal polynomials Theorem For orthonormal polynomials, there exist sequences of coefficients α k, k = 1, 2,... and β k, k = 1, 2,... such that βk+1 p k+1 (λ) = (λ α k+1 )p k (λ) β k p k 1 (λ), k = 0, 1,... (8) where p 1 (λ) 0, p 0 (λ) 1/ β 0, β 0 = α k+1 = λp k, p k, k = 0, 1,... and β k is computed such that p k = 1 b a dα

17 Relations between monic and orthonormal polynomials Assume that we have a system of monic polynomials p k satisfying a three-term recurrence (6), then we can obtain orthonormal polynomials ˆp k by normalization Using equation (6) p k+1 ˆp k+1 = After some manipulations ˆp k (λ) = p k(λ) p k, p k 1/2 ( λ p k λp ) k, p k ˆp k p k 2 p k p k 1 ˆp k 1 p k+1 p k ˆp k+1 = (λ λˆp k, ˆp k )ˆp k p k p k 1 ˆp k 1

18 Note that and λˆp k, ˆp k = λp k, p k p k 2 βk+1 = p k+1 p k Therefore the coefficients α k are the same and β k = γ k If we have the coefficients of monic orthogonal polynomials we just have to take the square root of γ k to obtain the coefficients of the corresponding orthonormal polynomials

19 Jacobi matrices If the orthonormal polynomials exist for all k, there is an infinite symmetric tridiagonal matrix J associated with them α 1 β1 β1 α 2 β2 J = β2 α 3 β Since it has positive subdiagonal elements, the matrix J is called an infinite Jacobi matrix Its leading principal submatrix of order k is denoted as J k Orthogonal polynomials are fully described by their Jacobi matrices

20 Christoffel Darboux relation Theorem Let p k, k = 0, 1,... be orthonormal polynomials, then k i=0 p i (λ)p i (µ) = p k+1 (λ)p k (µ) p k (λ)p k+1 (µ) β k+1, if λ µ λ µ k pi 2 (λ) = β k+1 [p k+1 (λ)p k(λ) p k (λ)p k+1(λ)] i=0 Corollary For monic orthogonal polynomials we have k i=0 γ k γ k 1 γ i+1 p i (λ)p i (µ) = p k+1(λ)p k (µ) p k (λ)p k+1 (µ), if λ µ λ µ (9)

21 Properties of zeros Let P k (λ) = ( p 0 (λ) p 1 (λ)... p k 1 (λ) ) T In matrix form, the three-term recurrence is written as λp k = J k P k + η k p k (λ)e k (10) where J k is the Jacobi matrix of order k and e k is the last column of the identity matrix (η k = β k ) Theorem The zeros θ (k) j of the orthonormal polynomial p k are the eigenvalues of the Jacobi matrix J k

22 Proof. If θ is a zero of p k, from equation (10) we have θp k (θ) = J k P k (θ) This shows that θ is an eigenvalue of J k and P k (θ) is a corresponding (unnormalized) eigenvector J k being a symmetric tridiagonal matrix, its eigenvalues (the zeros of the orthogonal polynomial p k ) are real and distinct Theorem The zeros of the orthogonal polynomials p k associated with the measure α on [a, b] are real, distinct and located in the interior of [a, b] see Szegö

23 Examples of orthogonal polynomials Jacobi polynomials dα(λ) = w(λ) dλ a = 1, b = 1, w(λ) = (1 λ) δ (1 + λ) β, δ, β > 1 Special cases: Chebyshev polynomials of the first kind: δ = β = 1/2 C k (λ) = cos(k arccos λ) They satisfy C 0 (λ) 1, C 1 (λ) λ, C k+1 (λ) = 2λC k (λ) C k 1 (λ)

24 The zeros of C k are ( ) 2j + 1 π λ j+1 = cos, j = 0, 1,... k 1 k 2 The polynomial C k has k + 1 extremas in [ 1, 1] ( ) jπ λ j = cos, j = 0, 1,..., k k and C k (λ j ) = ( 1)j For k 1, C k has a leading coefficient 2 k 1 0 i j < C i, C j > α = π 2 i = j 0 π i = j = 0

25 Chebyshev polynomials (first kind) C k, k = 1,..., 7 on [ 1.1, 1.1]

26 Let π 1 n = { poly. of degree n in λ whose value is 1 for λ = 0 } Chebyshev polynomials provide the solution of the minimization problem min max q n(λ) λ [a,b] q n π 1 n The solution is written as min q n π 1 n max q n(λ) = max λ [a,b] see Dahlquist and Björck λ [a,b] ( ) C 2λ (a+b) n b a ( ) C a+b n = b a 1 C n ( a+b b a )

27 Chebyshev polynomials of the second kind δ = β = 1/2 sin(k + 1)θ U k (λ) =, λ = cos θ sin θ They satisfy the same three term recurrence as the Chebyshev polynomials of the first kind but with initial conditions U 0 1, U 1 2λ Of all monic polynomials q k, 2 k U k gives the smallest L 1 norm q k 1 = 1 1 q k (λ) dλ

28 Chebyshev polynomials (second kind) U k, k = 1,..., 7 on [ 1.1, 1.1]

29 Legendre polynomials δ = β = 0 (k+1)p k+1 (λ) = (2k+1)λP k (λ) kp k 1 (λ), P 0 (λ) 1, P 1 (λ) λ The Legendre polynomial P k is bounded by 1 on [ 1, 1]

30 Legendre polynomials P k, k = 1,..., 7 on [ 1.1, 1.1]

31 Laguerre polynomials The interval is [0, ] and the weight function is e λ The recurrence relation is (k + 1)L k+1 = (2k + 1 λ)l k kl k 1 L 0 1, L 1 1 λ

32 Laguerre polynomials L k, k = 1,..., 7 on [ 2, 20]

33 Hermite polynomials The interval is [, ] and the weight function is e λ2 The recurrence relation is H k+1 = 2λH k 2kH k 1 H 0 1, H 1 2λ

34 Hermite polynomials H k, k = 1,..., 7 on [ 10, 10]

35 Variable-signed weight functions What happens if the measure α is not positive? Theorem Assume that all the moments exist and are finite For any k > 0, there exists a polynomial p k of degree at most k such that p k is orthogonal to all polynomials of degree k 1 with respect to w see G.W. Struble The important words in this result are: of degree at most k In some cases the polynomial p k can be of degree less than k

36 C(k) = set of polynomials of degree k orthogonal to all polynomials of degree k 1 C(k) is called degenerate if it contains polynomials of degree less than k If C(k) is non-degenerate it contains one unique polynomial (up to a multiplicative constant) Theorem Let C(k) be non-degenerate with a polynomial p k Assume C(k + n), n > 0 is the next non-degenerate set. Then p k is the unique (up to a multiplicative constant) polynomial of lowest degree in C(k + m), m = 1,..., n 1

37 d k d k 1 1 p k (λ) = (α k λ d k d k 1 + β k,i λ i )p k 1 (λ) γ k 1 p k 2 (λ), k = 2,. i=0 d 1 1 p 0 (λ) 1, p 1 (λ) = (α 1 λ d 1 + β 1,i λ i )p 0 (λ) i=0 (11) The coefficient of p k 1 contains powers of λ depending on the difference of the degrees of the polynomials in the non-degenerate cases The coefficients α k and γ k 1 have to be nonzero

38 Matrix orthogonal polynomials We would like to have matrices as coefficients of the polynomials For our purposes we just need 2 2 matrices Definition For λ real, a matrix polynomial p i (λ), which is a 2 2 matrix, is defined as p i (λ) = i j=0 λ j C (i) j where the coefficients C (i) j are given 2 2 real matrices If the leading coefficient is the identity matrix, the matrix polynomial is said to be monic The measure α(λ) is a matrix of order 2 that we suppose to be symmetric and positive semi definite

39 We assume that the (matrix) moments M k = exist for all k b a λ k dα(λ) (12) The inner product of two matrix polynomials p and q is defined as p, q = b a p(λ) dα(λ)q(λ) T (13)

40 Two matrix polynomials in a sequence p k, k = 0, 1,... are said to be orthonormal if < p i, p j >= δ i,j I 2 (14) where δ i,j is the Kronecker symbol and I 2 the identity matrix of order 2 Theorem Sequences of matrix orthogonal polynomials satisfy a block three term recurrence p j (λ)γ j = λp j 1 (λ) p j 1 (λ)ω j p j 2 (λ)γ T j 1 (15) p 0 (λ) I 2, p 1 (λ) 0 where Γ j, Ω j are 2 2 matrices and the matrices Ω j are symmetric

41 The block three-term recurrence can be written in matrix form as λ[p 0 (λ),..., p k 1 (λ)] = [p 0 (λ),..., p k 1 (λ)]j k + [0,..., 0, p k (λ)γ k ] (16) where Ω 1 Γ T 1 J k = Γ 1 Ω 2 Γ T Γ k 2 Ω k 1 Γ T k 1 Γ k 1 Ω k is a block tridiagonal matrix of order 2k with 2 2 blocks

42 Let P(λ) = [p 0 (λ),..., p k 1 (λ)] T J k P(λ) = λp(λ) [0,..., 0, p k (λ)γ k ] T Theorem For λ and µ real, we have the matrix analog of the Christoffel Darboux identity, k 1 (λ µ) p j (µ)pj T (λ) = p k 1 (µ)γ T k pt k (λ) p k(µ)γ k pk 1 T (λ) j=0 (17)

43 F.V. Atkinson, Discrete and continuous boundary problems, Academic Press, (1964) C. Brezinski, Biorthogonality and its applications to numerical analysis, Marcel Dekker, (1992) T.S. Chihara, An introduction to orhogonal polynomials, Gordon and Breach, (1978) G. Dahlquist and A. Björck, Numerical methods in scientific computing, volume I, SIAM, (2008) G. Dahlquist, S.C. Eisenstat and G.H. Golub, Bounds for the error of linear systems of equations using the theory of moments, J. Math. Anal. Appl., v 37, (1972), pp G. Dahlquist, G.H. Golub and S.G. Nash, Bounds for the error in linear systems. In Proc. of the Workshop on Semi Infinite Programming, R. Hettich Ed., Springer (1978), pp

44 W. Gautschi, Orthogonal polynomials: computation and approximation, Oxford University Press, (2004) G.W. Struble, Orthogonal polynomials: variable signed weight functions, Numer. Math., v 5, (1963), pp G. Szegö, Orthogonal polynomials, Third Edition, American Mathematical Society, (1974)