Hermite Interpolation and Sobolev Orthogonality

Transcription

1 Acta Applicandae Mathematicae 61: 87 99, Kluwer Academic Publishers Printed in the Netherlands 87 Hermite Interpolation and Sobolev Orthogonality ESTHER M GARCÍA-CABALLERO 1,, TERESA E PÉREZ 2, and MIGUEL A PIÑAR 2, 1 Departamento de Matemáticas, Universidad de Jaén, Jaén, Spain emgarcia@ujaenes 2 Departamento de Matemática Aplicada and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Spain {tperez, mpinar}@goliatugres (Received: 30 June 1999) Abstract In this paper, we study orthogonal polynomials with respect to the bilinear form where (f, g) S = V(f)AV(g) T + u, f (N) g (N), V(f)= (f (c 0 ), f (c 0 ),,f (n 0 1) (c0 ),,f(c p ), f (c p ),,f (n p 1) (cp )), u is a regular linear functional on the linear space P of real polynomials, c 0,c 1,,c p are distinct real numbers, n 0,n 1,,n p are positive integer numbers, N = n 0 +n 1 + +n p,anda is a N N real matrix with all its principal submatrices nonsingular We establish relations with the theory of interpolation and approximation Mathematics Subject Classifications (2000): 33C45, 42C05 Key words: Sobolev bilinear forms, orthogonal polynomials, Hermite interpolation 1 Introduction Sobolev orthogonality has been studied for years For different families of polynomials, there exist several results about recurrence relations, asymptotics, algebraic and differentation properties, zeros, etc (see, for instance, Alfaro et al (1999), Jung et al (1997), Kwon and Littlejohn (1995, 1998), Marcellán et al (1996), Pérez and Piñar (1996)); but there exist very few results establishing the relation with the theory of interpolation and approximation In this paper, we study a connection between the general Hermite interpolation problem and a kind of discrete-continuous Sobolev bilinear form We consider the bilinear form (f, g) S = V(f)AV(g) T + u, f (N) g (N), (1) Research supported by Junta de Andalucía, G I FQM 0178 Research partially supported by Junta de Andalucía, GI FQM 0229, DGES PB and INTAS ext

2 88 E M GARCÍA-CABALLERO ET AL where V(f) = (f (c 0 ), f (c 0 ),,f (n0 1) (c 0 ),, f (c p ), f (c p ),,f (np 1) (c p )), u is a regular linear functional on the linear space P of real polynomials, c 0,c 1,, c p are distinct real numbers, n 0,n 1,,n p are positive integer numbers, N = n 0 + n 1 + +n p,andais an N N real matrix with all its principal submatrices nonsingular Since the expression (1) involves derivatives, this bilinear form is non-standard, and by analogy with the usual terminology we call it a discrete-continuous Sobolev bilinear form The structure of the paper is as follows: in Section 2, we give a description of the monic polynomials {Q n } n which are orthogonal with respect to (1) If we denote by {P n } n the monic orthogonal polynomials with respect to the functional u, then {Q n } N 1 n=0 are orthogonal with respect to the discrete part of the symmetric bilinear form (1), and if n N,wehave and Q (k) n (c i) = 0, i = 0, 1,,p, k = 0, 1,,n i 1, Q (N) n (x) = n! (n N)! P n N (x) In Section 3 we study the kernel polynomials L n (x, y) corresponding to {Q n } n We will show that L (N,N) n (x, y) = 2N x N y L n(x, y) = K N n N (x, y), n N, where K n (x, y), n 0, are the kernel polynomials corresponding to {P n } n In Section 4, we show that the polynomials {Q n } n are the error terms in an interpolation problem In Section 5, when the Sobolev bilinear form becomes an inner product we give a relation between the Sobolev orthogonality and a problem of simultaneous polynomial interpolation and approximation Finally, in Section 6, we present some interesting particular cases of (1) and some examples of monic orthogonal polynomial sequences which are orthogonal with respect to the bilinear form (1), using Laguerre and Jacobi functionals 2 The Sobolev Discrete-Continuous Bilinear Form We define a symmetric bilinear form on the linear space of real polynomials, P, in the following way (f, g) S = V(f)AV(g) T + u, f (N) g (N), (2)

3 HERMITE INTERPOLATION AND SOBOLEV ORTHOGONALITY 89 where V(f) = (f (c 0 ), f (c 0 ),,f (n0 1) (c 0 ),, f (c p ), f (c p ),,f (np 1) (c p )), u is a regular linear functional on P (see Chihara (1978)), c 0,c 1,,c p are distinct real numbers, n 0,n 1,,n p are positive integers, N = n 0 + n 1 + +n p,anda is a N N real matrix such that each of its principal submatrices are nonsingular Set p w N (x) = (x c i ) n i In P, we can consider the basis given by { {Lik B = (x) },1,,p, { w N (x)x } } j k=0,1,,n i 1 j 0 The polynomials L ik P N 1 are the generalized Lagrange polynomials (see Stoer and Burlirsch (1993)) They are defined as follows Starting with the auxiliary polynomials l ik (x) = (x c i) k p ( ) x nj cj, put k! j i c i c j i = 0, 1,,p; k = 0, 1,,n i 1, L i,ni 1(x) := l i,ni 1(x), i = 0, 1,,p, and recursively for k = n i 2,n i 3,,0, L ik (x) := l ik (x) i 1 v=k+1 l (v) ik (c i)l iv (x) The essential characteristic of these polynomials is that they satisfy the relations: L (j) ik (c v) = δ iv δ kj, i,v = 0, 1,,p; k,j = 0, 1,,n i For n N 1, the associated Gram matrix G n is given by the nth order principal submatrix of the matrix AForn N, the associated Gram matrix is given by ( ) A 0 G n =, 0 B n N where B n N is the Gram matrix associated with the regular linear functional u in the basis B ={D (N) [w N (x)x j ],j 0} The existence of a sequence of monic

4 90 E M GARCÍA-CABALLERO ET AL polynomials, {Q n } n, which is orthogonal with respect to (2) is assured because the associated Gram matrix is nonsingular and therefore, the discrete-continuous Sobolev bilinear form (2) is regular These polynomials will be called Sobolev orthogonal polynomials THEOREM 1 Let {Q n } n be the MOPS with respect to the Sobolev discrete-continuous form (2) and let {P n } n be the MOPS associated with the regular linear functional u (i) The polynomials {Q n } N 1 n=0 are orthogonal with respect to the discrete bilinear form (f, g) D = V(f)AV(g) T (3) (ii) If n N, then Q (k) n (c i) = 0, i = 0, 1,,p; k = 0, 1,,n i 1, (4) Q (N) n! n (x) = (n N)! P n N(x) (5) Proof (i) If 0 n, m < N,thenQ (N) n (x) = Q (N) m (x) = 0, and obviously (Q n,q m ) S = (Q n,q m ) D (ii) For n N, from the orthogonality of the polynomial Q n, we deduce 0 = (Q n,l ik ) S = (Q n,l ik ) D = V(Q n )AV(L ik ) T = V(Q n )A(0,,1,,0) T, for 0 i p, 0 k n i 1 Thus, the vector V(Q n ) is the only solution of a homogeneous linear system with N equations and N unknows, whose coefficient matrix A is nonsingular Then we conclude Q (k) n (c i) = 0, i = 0, 1,,p, k = 0, 1,,n i 1, that is, Q n contains the factor (x c 0 ) n 0(x c 1 ) n 1 (x c p ) n p In this way, if n, m N (Q n,q m ) S = u, Q (N) n Q (N) m = k n δ n,m, k n 0 That is, the polynomials {Q (N) n } n N are orthogonal with respect to the linear functional u, and equality (5) follows from a simple inspection of the leading coefficients Reciprocally, we are going to show that a system of monic polynomials {Q n } n satisfying Equations (4) and (5) is orthogonal with respect to some discrete-continuous Sobolev form like (2) This result could be considered as a Favard-type theorem THEOREM 2 Let {P n } n be the MOPS associated with a regular linear functional u and N 1 be a given integer number Let {Q n } n be a sequence of monic polynomials satisfying (i) deg Q n = n, n = 0, 1,,

5 HERMITE INTERPOLATION AND SOBOLEV ORTHOGONALITY 91 (ii) Q (k) n (c i) = 0, 0 i p, 0 k n i 1, n N, (iii) Q (N) n (x) = n! (n N)! P n N (x), n N Then, there exists a nonsingular and symmetric real matrix A, ofordern, such that {Q n } n is the monic orthogonal polynomial sequence associated with the Sobolev bilinear form defined by (2) Proof Obviously the polynomial Q n, with n N, is orthogonal to every polynomial of degree less than or equal to n 1 with respect to a Sobolev bilinear form like (2), containing an arbitrary matrix A in the discrete part and the functional u in the second part Next, we will show that we can recover the matrix A from the N first polynomials Q k, k = 0, 1,,N 1 Let us denote by Q = V(Q 0 ) V(Q 1 ) V(Q N 1 ) The matrix Q is nonsingular since it can be expressed as the product of two nonsingular matrices To this end, let us expand the polynomials {Q n } n in powers of x Q n (x) = p n,j x j, p n,n = 1, n 0 (6) Let P be the nonsingular matrix obtained from the coefficients p n,j,for0 j n and 0 n N 1, p 0,0 0 0 p 1,0 p 1,1 0 P = p N 1,0 p N 1,1 p N 1,N 1 Let us denote by H i (x), the matrices obtained from the first n i 1 derivatives of the Hamel basis of the linear space P N 1 where i = 0, 1,,p;thatis H i (x) = ( D k (x j ) ),1,,N 1, k=0,1,,n i 1 and denote by V the square matrix obtained from the matrices H i (x), i = 0, 1,,pas follows V(x) = ( H 0 (c 0 ) H 1 (c 1 ) H p (c p ) ) Therefore V is a nonsingular matrix From Equation (6) we deduce Q = PV, and we conclude the nonsingularity of Q

6 92 E M GARCÍA-CABALLERO ET AL Let D be a diagonal nonsingular matrix Define A = Q 1 D(Q 1 ) T (7) Obviously A is symmetric and nonsingular and since QAQ T = D, the polynomials Q 0,Q 1,,Q N 1 are orthogonal with respect to the bilinear form (2), with the matrix A in the discrete part Besides, the elements in the diagonal of D are the values (Q k,q k ) S for k = 0, 1,,N 1 Remark Observe that the matrix A is not unique, because its construction depends on the arbitrary nonsingular matrix D 3 Kernel Polynomials Let K n (x, y) = P i (x)p i (y) u, Pi 2, n 0, be the kernel polynomials corresponding to {P n } n,andlet L n (x, y) = Q i (x)q i (y) (Q i,q i ) S, n 0, be the kernel polynomials corresponding to {Q n } n The next theorem shows the relation between K n (x, y) and L n (x, y) THEOREM 3 In the above hypothesis, L (N,N) n (x, y) = 2N x N y N L n(x, y) = K n N (x, y), n N (8) Proof From Theorem 1 we can assure the following equalities L n (x, y) = = N 1 N 1 Q i (x)q i (y) (Q i,q i ) S + Q i (x)q i (y) (Q i,q i ) D + Q i (x)q i (y) (Q i,q i ) S Q i (x)q i (y) u, (Q (N) i ) 2

7 HERMITE INTERPOLATION AND SOBOLEV ORTHOGONALITY 93 Taking derivatives with respect to both variables, we get L (N,N) Q (N) i n (x, y) = = 2N x N y N L n(x, y) = n N P i N (x)p i N (y) u, Pi N 2 = (x)q (N) i (y) u, (Q (N) i ) 2 P j (x)p j (y) u, P 2 j = K n N (x, y) From Theorem 2 we know that we can recover the matrix A from the N first polynomials Q k, k = 0, 1,,N 1, in fact, from (7), A = Q 1 D(Q 1 ) T, therefore A 1 = Q T D 1 Q As an immediate consequence we have the following result THEOREM 4 A 1 = (L k,l ) k=0,1,,p, l=0,1,,p where L k,l = ( L (i,j) N 1 (c k,c l ) ),1,,n k 1,1,n l 1 COROLLARY 1 (Reproducing property) (f (t), L (0,k) n (t, c i )) S = f (k) (c i ), i = 0, 1,,p, k = 0, 1,,n i 1, n N 1 Proof From the previous theorem and (4) we have (f (t), L (0,k) n (t, c i )) S = (f (t), L (0,k) N 1 (t, c i)) S = V(f)AV(L (0,k) N 1 )T = f (k) (c i ), i = 0, 1,,p, k = 0, 1,,n i 1, n N 1 4 Sobolev Orthogonal Polynomials and Interpolation In this section we will show that the polynomials {Q n } n can be expressed as the interpolation error of a Nth primitive of {P n N } n N THEOREM 5 In the above hypothesis, let {R n } n N be a sequence of Nth monic primitives of the polynomials {P n N } n N then Q n (x) = R n (x) p i 1 R n (j) (c i)l ij (x), n N,

8 94 E M GARCÍA-CABALLERO ET AL ie, Q n (x) for n N is the error for the Hermite interpolation polynomial for R n Proof Integrating in (5) N times, we get Q n (x) = R n (x) + p i 1 A ij L ij (x), n N Using (4), we deduce A ij = R n (j) i), i = 0,,p, k = 0, 1,,n i 1 Reciprocally, we have THEOREM 6 Let {R n } n be a sequence of monic polynomials such that deg R n = n, n = 0, 1,Let {Q n } n be the sequence of polynomials given by Q n (x) = R n (x), n = 0, 1,,N 1, (9) p i 1 Q n (x) = R n (x) R n (j) (c i)l ij (x), n N, (10) where c 0,c 1,,c p are distinct real numbers If {R n (N) } n N is an orthogonal polynomial sequence with respect to some regular linear functional u, then there exists a nonsingular and symmetric real matrix A, of order N, such that {Q n } n N is the MOPS associated with the Sobolev bilinear form defined by (2) Proof By (9) and (10) we have deg Q n = deg R n = n and for n N we have Q (k) n Moreover, (c i) = R (k) (c i) Q (N) n (x) = R n (N) (x) = n p i 1 R n (j) (c i)l (k) ij (c i) = 0 n! (n N)! P n N(x), n N, where {P n } n is the MOPS associated with u From Theorem 2 we conclude that {Q n } n is the MOPS with respect to the bilinear form defined by (2), where A = R 1 D(R 1 ) T, R is the matrix R = V(R 0 ) V(R 1 ) V(R N 1 ) = V(Q 0 ) V(Q 1 ) V(Q N 1 ), and D is an arbitrary nonsingular diagonal matrix

9 HERMITE INTERPOLATION AND SOBOLEV ORTHOGONALITY 95 5 Sobolev Orthogonal Polynomials and Approximation Let us assume that u is positive definite and A is a positive definite, symmetric and real matrix In this case (2) is an inner product If u is positive definite, then there exists a positive definite Borel measure µ satisfying u, f = f(x)dµ(x) R (see Chihara (1978), p 57), and in this case the discrete-continuous Sobolev inner product (2) can be written as (f, g) S = V(f)AV(g) T + f (N) (x)g (N) (x) dµ(x) (11) R Let I the convex hull of the set supp(µ) {c i } p LetW 2 N Sobolev space: [I,dµ] denote the W N 2 [I,dµ] ={f : I R; f CN 1 (I), f (N) L 2 µ (I)} The equation f S = (f, f ) S defines a norm in W2 N[I,dµ], andw 2 N [I,dµ] becomes a normed linear space (see Davis (1975), p 160) This space is strictly conv/ex (see Davis (1975), p 141) In the normed linear space W2 N [I,dµ] with the strictly convex norm S the problem of best approximation has a unique solution We want to compute the best approximation of f W2 N[I,dµ] in P n THEOREM 7 Let f W2 N [I,dµ] Letv be the best approximation of f in (P n,(, ) S ) and let w be the best approximation of f (N) in (P n N, dµ), then (i) v (k) (c i ) = f (k) (c i ), i = 0, 1,,p, k = 0, 1,,n i 1 (ii) v (N) (x) = w(x) Proof We know that v = (f, Q i ) S Q i 2 Q i, S where (f, Q i ) S / Q i 2 S are known as the Fourier coefficients of v Therefore v(x) = (f (t), L n (t, x)) S (i) Using Corollary 1 v (k) (c i ) = (f (t), i = 0, 1,,p, k = 0, 1,,n i 1 L (0,k) n (t, c i )) S = (f (t), L (0,k) N 1 (t, c i)) S = f (k) (c i ),

10 96 E M GARCÍA-CABALLERO ET AL (ii) Using (5) v (N) (x) = (f (t), L (0,N) n (t, x)) S = (f, Q i ) S Q i 2 S Q (N) i (x) u, f (N) Q (N) n N i u, f (N) P i = u, (Q (N) i ) 2 Q(N) i (x) = u, P 2 i P i (x) = w(x) Let {Q n } n be the MOPS with respect to the Sobolev discrete-continuous inner product (11) Let v be the best approximation of f W N 2 [I,dµ] in (P n,(, ) S ) THEOREM 8 In the above hypothesis: (i) If n N 1, we have that (f, Q i ) D v = Q i 2 Q i D (ii) If n N, N 1 (f, Q i ) D (i N)! u, f (N) P i N v = Q i 2 Q i + D i! u, Pi N 2 Q i 6 Particular Cases and Examples 61 PARTICULAR CASES 611 Classical Polynomial Interpolation (Lagrange Interpolation) If n 0 = n 1 = = n p = 1wehavethatN = p, so the discrete-continuous Sobolev bilinear form (2) can be written as g(c 0 ) g(c 1 ) (f, g) S = (f (c 0 ), f (c 1 ),, f (c N 1 ))A + u, f (N) g (N) g(c N 1 ) In this case we have the connection with the classical polynomial interpolation problem 612 Taylor Interpolation If p = 0wehavethatN = n 0, so the discrete-continuous Sobolev bilinear form (2) can be written as

11 HERMITE INTERPOLATION AND SOBOLEV ORTHOGONALITY 97 (f, g) S = (f (c 0 ), f (c 0 ),,f (N 1) (c 0 ))A g(c 0 ) g (c 0 ) g (N 1) (c 0 ) + u, f (N) g (N) In this case we have the connection with the Taylor interpolation problem 62 EXAMPLES 621 Laguerre Case Let α R, monic generalized Laguerre polynomials are defined in Szegő (1975), p 102, by means of their explicit representation L (α) n (x) = ( 1) j ( ) n + α ( 1)n n! x j! n j j, n 0, where ( a k ) denotes the generalized binomial coefficient ( a k ) = (a k + 1) k k! and (a k + 1) k stands for the so-called Pochhammer s symbol defined by (b) 0 = 1, (b) n = b(b + 1) (b + n 1), b R, n 0 When α is not a negative integer, Laguerre polynomials are orthogonal with respect to a regular linear functional u (α) This linear functional is positive definite for α> 1 We know that the derivatives of Laguerre polynomials are again Laguerre polynomials d (x) = nl(α+1) (x), n 1 dx L(α) n n 1 Let {Q n } n be the sequence of monic polynomials given by Q n (x) = L (α N) n (x), n = 0, 1,,N 1, (12) p i 1 Q n (x) = L (α N) ( ) n (x) L (α N) (j)(ci n )L ij (x), n N, (13) where L ij (x), i = 0,,N 1, are the generalized Lagrange polynomials From Theorem 1, we conclude that the sequence {Q n } n is orthogonal with respect to the Sobolev bilinear form (f, g) S = V(f)AV(g) T + u (α),f (N) g (N),

12 98 E M GARCÍA-CABALLERO ET AL where c 0,c 1,,c p are distinct real numbers, n 0,n 1,,n p are positive integer numbers, N = n 0 + n 1 + +n p, the matrix A is given by A = Q 1 D(Q 1 ) T, Q is the matrix given by V(L (α N) 0 ) V(L (α N) Q = 1 ) V(L (α N) N 1 ) and D is an arbitrary nonsingular diagonal matrix We observe that Q n,n 0, given by (12) and (13), is the error for the Hermite interpolation polynomial for Laguerre polynomials L (α N) n Remark Laguerre polynomials {L ( N) n } n contain for n N, the factor x N Therefore, for α = 0andc 0 = 0, Theorem 2 provides Sobolev orthogonality for these polynomials 622 Jacobi Case For α and β real numbers, the generalized Jacobi polynomials can be defined (Szegő (1975), p 68) by means of their explicit representation ( )( )( ) P n (α,β) n + α n + β x 1 n m ( ) x + 1 m (x) =, n 0 m n m 2 2 m=0 When α, β and α + β + 1 are not a negative integers, Jacobi polynomials are orthogonal with respect to a regular linear functional u (α,β) This linear functional is positive definite for α, β> 1 Set P n (x), n 0, the monic Jacobi polynomials We know that the derivatives of Jacobi polynomials are again Jacobi polynomials d P dx n (α,β) (x) = np n 1 (x), n 1 Let {Q n } n be the sequence of monic polynomials given by Q n (x) = P n (α N,β N) (x), n = 0, 1,,N 1, (14) and, for n N, Q n (x) = P n (α N,β N) (x) p i 1 ( P (α N,β N) n where α, β and α + β 2N + 1 are not negative integers ) (j) (ci )L ij (x), (15)

13 HERMITE INTERPOLATION AND SOBOLEV ORTHOGONALITY 99 From Theorem 1, we conclude that the sequence {Q n } n is orthogonal with respect to the Sobolev bilinear form (f, g) S = V(f)AV(g) T + u (α,β),f (N) g (N), where c 0,c 1,,c p are distinct real numbers, n 0,n 1,,n p are positive integer numbers, N = n 0 + n 1 + +n p, the matrix A is given by A = Q 1 D(Q 1 ) T, Q is the matrix given by V( P (α N,β N) 0 ) V( P (α N,β N) Q = 1 ) V( P (α N,β N) N 1 ) and D is an arbitrary nonsingular diagonal matrix We observe that Q n,n 0, given by (14) and (15), is the error for the Hermite interpolation polynomial for Jacobi polynomials P n (α N,β N) Remark Jacobi polynomials { P n ( 1, 1) } n contain for n 2, the factor x 2 1 Therefore, for α = β = 1, n 0 = n 1 = 1andc 0 = 1, c 1 = 1, Theorem 2 provides Sobolev orthogonality for these polynomials (see Kwon and Littlejohn (1998)) For α = β = n 0, n 0 = n 1 and c 0 = 1, c 1 = 1, Theorem 2 provides Sobolev orthogonality for { P ( n 0, n 0 ) n } n References Alfaro, M, Pérez, T E, Piñar, M A and Rezola, M L (1999) Sobolev orthogonal polynomials: The discrete-continuous case, Methods Appl Anal 6(4), Chihara, T S (1978) An Introduction to Orthogonal Polynomials, Gordon and Breach, New York Davis, P J (1975) Interpolation and Approximation, Dover Publications, New York Jung, I H, Kwon, K H and Lee, J K (1997) Sobolev orthogonal polynomials relative to λp(c)q(c) + τ,p (x)q (x), Comm Korea Math Soc 12, Kwon, K H and Littlejohn, L L (1995) The orthogonality of the Laguerre polynomials {L ( k) n (x)} for positive integers k, Ann Numer Math 2, Kwon, K H and Littlejohn, L L (1998) Sobolev orthogonal polynomials and second-order differential equations, Rocky Mountain J Math 28, Marcellán, F, Pérez, T E, Piñar, M A and Ronveaux, A (1996) General Sobolev orthogonal polynomials, J Math Anal Appl 200, Pérez, T E and Piñar, M A (1996) On Sobolev orthogonality for the generalized Laguerre polynomials, J Approx Theory 86, Stoer, J and Burlirsch, R (1993) Introduction to Numerical Analysis, 2nd edn, Springer, New York Szegő, G (1975) Orthogonal Polynomials, 4th edn, Amer Math Soc Colloq Publ 23, Amer Math Soc, Providence, RI