70 3 Orthogonal Polynomials and Related Approximation Results 3.2 Jacobi Polynomials

3.2.1 Basic Properties

α,β The Jacobi polynomials, denoted byJ n (x), are orthogonal with respect to the Jacobi weight function ωα,β (x):= (1 x) α (1+x) β overI:= ( 1,1), namely, − − 1 α,β α,β α,β α,β Jn (x)Jm (x)ω (x)dx= γn δmn, (3.88) 1 �− α,β α,β 2 α,β 1 where γn = J n . The weight function ω belongs toL (I) if and onlyif � �ωα,β α,β > 1 (to be assumed throughout this section). −α,β α,β Letk n be the leading coefficient ofJ n (x). According to Theorem 3.1, there α,β α,β exists a unique sequence of monic orthogonal polynomials J n (x)/kn . This class of Jacobi weight functions leads to Jacobi{ polynomials with many attractive properties that are not shared by general orthogonal polynomials.�

3.2.1.1 Sturm-Liouville Equation

We first show that the Jacobi polynomialsare the eigenfunctionsof a singular Sturm- Liouville operator defined by

α β α+1 β +1 L u:= (1 x) − (1+x) − ∂x (1 x) (1+x) ∂xu(x) α,β − − − (3.89) 2 2 = (x 1) ∂ u(x)+ α β + (α + β +2)x ∂xu(x). − x − � � More precisely, we have � �

Theorem 3.16. The Jacobi polynomials are the eigenfunctions of the singular Sturm-Liouville problem:

α,β α,β α,β Lα,β Jn (x)= λn Jn (x), (3.90) and the corresponding eigenvalues are

α,β λn =n(n+ α + β +1). (3.91)

Proof. For anyu P n, we haveL u P n. Using integration by parts twice, we ∈ α,β ∈ find that for any φ P n 1, ∈ − L α,β α,β α,β L (3.88) α,β Jn ,φ ωα,β = ∂xJn ,∂xφ ωα+1,β+1 = Jn , α,β φ ωα,β =0.

�L α,β � � � � � Since α,β Jn P n, the uniqueness of orthogonal polynomials implies that there ∈ α,β exists a constant λn such that

α,β α,β α,β Lα,β Jn = λn Jn . 3.2 Jacobi Polynomials 71

α,β n To determine λn , we compare the coefficient of the leading termx on both sides, and find α,β α,β α,β kn n(n+ α + β +1)=k n λn , α,β α,β α,β wherek n is the leading coefficient ofJ n . Hence, we have λn =n(n+ α+ β +1). �� Remark 3.2. Observe from integration by parts that the Sturm-Liouville operator

L is self-adjoint with respect to the inner product( , ) α,β , i.e., α,β · · ω L L ( α,β φ,ψ)ωα,β = (φ, α,β ψ)ωα,β , (3.92) for any φ,ψ u:L u L 2 (I) . ∈ α,β ∈ ωα,β As pointed out� in Theorem 4.2.2 of� Szego¨ (1975), the differential equation L α,β u= λu,

has a polynomial solution not identically zero if and only if λ has the formn(n+ α,β α + β +1). This solution isJ n (x) (up to a constant), and no solution which is α,β linearly independent ofJ n (x) can be a polynomial. Moreover, we can show that n α,β n k Jn (x)= ∑ ak (x 1) , k=0 − where n α,β ak+1 γn k(k+ α + β +1) n = − . (3.93) ak 2(k+1)(k+ α +1) Assume that the Jacobi polynomials are normalized such that

n+ α Γ (n+ α +1) an =J α,β (1)= = , (3.94) 0 n n n!Γ (α +1) � � where Γ ( ) is the Gamma function (cf. Appendix A). We can derive from (3.93) the leading coefficient· Γ (2n+ α + β +1) an =k α,β = . (3.95) n n 2nn!Γ (n+ α + β +1) Moreover, working out a n byusing (3.93), we find { k}

n k α,β Γ (n+ α +1) n Γ (n+k+ α + β +1) x 1 Jn (x)= ∑ − . (3.96) n!Γ (n+ α + β +1) k 0 k Γ (k+ α +1) 2 = � � � � α,β A direct consequence of Theorem 3.16 is the orthogonality of ∂xJn . Corollary 3.5. � �

1 α,β α,β α+1,β +1 α,β α,β ∂xJn ∂xJm ω dx= λn γn δnm. (3.97) 1 �− 72 3 Orthogonal Polynomials and Related Approximation Results

α,β Proof. Using integration by parts, Theorem 3.16 and the orthogonality of Jn , we obtain � � α,β α,β α,β α,β (3.90) α,β α,β 2 ∂xJ , ∂xJ = J ,L J = λ J δnm. n m ωα+1,β+1 n α,β m ωα,β n � n �ωα,β This� ends the proof.� � � �� α,β α+1,β +1 Since ∂xJn is orthogonal with respect to the weight ω , by Theorem α{,β } α+1,β +1 3.1, ∂xJn must be proportional toJ n 1 , namely, − α,β α,β α+1,β +1 ∂xJn (x)= μn Jn 1 (x). (3.98) − Comparing the leading coefficients on both sides leads to the proportionality con- stant: α,β nkn (3.95) 1 μα,β = = (n+ α + β +1). (3.99) n α+1,β +1 2 kn 1 − This gives the following important derivative relation:

α,β 1 α+1,β +1 ∂xJn (x)= (n+ α + β +1)J n 1 (x). (3.100) 2 − Applying this formula recursively yields

k α,β α,β α+k,β +k ∂x Jn (x)=d n,k Jn k (x),n k, (3.101) − ≥ where Γ (n+k+ α + β +1) dα,β = . (3.102) n,k 2kΓ (n+ α + β +1)

3.2.1.2 Rodrigues’ Formula

The Rodrigues’ formula for the Jacobi polynomials is stated below. Theorem 3.17. ( 1)n dn (1 x) α (1+x) β Jα,β (x)= − (1 x) n+α(1+x) n+β . (3.103) − n 2nn! dxn − � � Proof. For any φ Pn 1, using integration by parts leads to ∈ − 1 n n+α n+β ∂x (1 x) (1+x) φdx=... 1 − �− � 1 � n n+α n+β n = ( 1) (1 x) (1+x) ∂x φdx=0. − 1 − �− � � 3.2 Jacobi Polynomials 73

Hence, by Theorem 3.1, there exists a constantc n such that

n n+α n+β α β α,β ∂ (1 x) (1+x) =c n(1 x) (1+x) J (x). (3.104) x − − n � � Lettingx 1 andusing ( 3.94) leads to →

1 1 n n+α n+β cn = ∂x (1 x) (1+x) Jα,β (1) (1 x) α(1+x) β − x=1 n � − � ��� n n � = ( 1) n!2 . � − The proof is complete. �� We now present some consequences of the Rodrigues’ formula. First, expanding thenth-order derivative in ( 3.103) yields the explicit formula

n α,β n n+ α n+ β n j j J (x)=2 − (x 1) − (x+1) . (3.105) n ∑ j n j − j=0 � �� − � Second, replacingxby x in ( 3.103) immediately leads to the symmetric relation − Jα,β ( x)=( 1)nJβ ,α (x). (3.106) n − − n α,α Therefore, the special Jacobi polynomialJ n (x) (up to a constant, is referred to as the Gegenbauer or ultra-spherical polynomial), is an odd function for oddn and an even function for evenn. Moreover, using ( 3.94) and (3.106) leads to

Γ (n+ β +1) Jα,β ( 1)=( 1)n , (3.107) n − − n!Γ (β +1)

and by the Stirling’s formula (A.7),

Jα,β (1) n α and J α,β ( 1) n β forn 1. (3.108) n ∼ | n − | ∼ � As another consequence of (3.103), we derive the explicit formula of the normaliza- α,β tion constant γn in (3.88).

Corollary 3.6.

1 2 α,β α,β α,β Jn (x) ω (x)dx= γn 1 �− � � (3.109) 2α+β +1Γ (n+ α +1)Γ (n+ β +1) = . (2n+ α + β +1)n! Γ (n+ α + β +1) 74 3 Orthogonal Polynomials and Related Approximation Results

α,β Proof. Multiplying (3.103) byJ n and integrating the resulting equality over ( 1,1), we derive from integration by parts that − 1 α β α,β 2 (1 x) (1+x) [Jn (x)] dx 1 − �− n 1 ( 1) n n+α n+β α,β = −n ∂x (1 x) (1+x) Jn (x)dx 2 n! 1 − �− � � 2n 1 ( 1) n+α n+β n α,β = −n (1 x) (1+x) ∂x Jn (x)dx 2 n! 1 − �− α,β 1 kn n+α n+β = n (1 x) (1+x) dx 2 1 − �− (3.95) 2α+β +1Γ (n+ α +1)Γ (n+ β +1) = . (A.6) (2n+ α + β +1)n! Γ (n+ α + β +1)

This ends the proof. ��

3.2.1.3 Recurrence Formulas

The Jacobi polynomials are generated by the three-term recurrence relation:

α,β α,β α,β α,β α,β α,β Jn+1(x)= an x b n Jn (x) c n Jn 1(x),n 1, − − − ≥ 1 1 (3.110) Jα,β (x)=1�,J α,β (x)=� (α + β +2)x+ (α β), 0 1 2 2 −

where (2n+ α + β +1)(2n+ α + β +2) aα,β = , (3.111a) n 2(n+1)(n+ α + β +1) (β 2 α2)(2n+ α + β +1) bα,β = − , (3.111b) n 2(n+1)(n+ α + β +1)(2n+ α + β) (n+ α)(n+ β)(2n+ α + β +2) cα,β = . (3.111c) n (n+1)(n+ α + β +1)(2n+ α + β)

This relation allows us to evaluate the Jacobi polynomials at any given abscissa x [ 1,1], and it is the starting point to derive other properties. ∈Next,− we state several useful recurrence formulas involving different pairs of (α,β).

α+1,β Theorem 3.18. The Jacobi polynomial Jn (x) is a linear combination of α,β Jl (x),l=0,1,...,n, i.e., 3.2 Jacobi Polynomials 75

Γ (n+ β +1) Jα+1,β (x)= n Γ (n+ α + β +2) × n (3.112) (2l+ α + β +1)Γ (l+ α + β +1) α,β ∑ Jl (x). l=0 Γ (l+ β +1) Proof. In the Jacobi case, the kernel polynomial (3.17) takes the form

n 1 K x y Jα,β x Jα,β y (3.113) n( , )= ∑ α,β l ( ) l ( ). l=0 γl

α+1,β By Lemma 3.2, K n(x,1) are orthogonal with respect to ω . By the unique- { } ness of orthogonal polynomials (cf. Theorem 3.1),K n(x,1) must be proportional to α+1,β Jn , i.e., n Jα,β (1) K (x,1)= l Jα,β (x)=d α,β Jα+1,β (x). (3.114) n ∑ α,β l n n l=0 γl α,β The proportionality constantd n is determined by comparing the leading coeffi- cients of both sides of (3.114) and working out the constants, namely,

kα,β Jα,β (1) Γ (n+ α + β +2) dα,β = n n =2 α β 1 . n α+1,β α,β − − − kn γn Γ (α +1)Γ (n+ β +1) Inserting this constant into (3.114), we obtain (3.112) directly from (3.94) and (3.109). �� Remark 3.3. Thanks to( 3.106), it follows from( 3.112) that

Γ (n+ α +1) Jα,β +1(x)= n Γ (n+ α + β +2) × n (3.115) n l (2l+ α + β +1)Γ (l+ α + β +1) α,β ∑( 1) − Jl (x). l=0 − Γ (l+ α +1) Theorem 3.19. The Jacobi polynomials satisfy

α,β α,β 2 (n+ α +1)J n (n+1)J Jα+1,β = − n+1 , (3.116a) n 2n+ α + β +2 1 x − α,β α,β 2 (n+ β +1)J n + (n+1)J Jα,β +1 = n+1 . (3.116b) n 2n+ α + β +2 1+x Proof. In the Jacobi case, the Christoffel-Darboux formula (3.15) reads

α,β α,β α,β α,β α,β kn Jn+1(x)Jn (y) J n (x)Jn+1(y) Kn(x,y)= − , (3.117) kα,β α,β x y n+1γn − 76 3 Orthogonal Polynomials and Related Approximation Results

which, together with (3.114), leads to 1 Jα+1,β x K x 1 n ( )= α,β n( , ) dn α,β α,β α,β α,β α,β kn J (x)Jn (1) J n (x)J (1) = n+1 − n+1 . dα,β kα,β α,β x 1 n n+1γn − Working out the constants yields (3.116a). Replacingx in ( 3.116a) by x and using the symmetric property ( 3.106), we derive (3.116b) immediately. − �� We state below two useful formulas and leave their derivation as an excise (see Problem 3.7). Theorem 3.20.

α,β α,β 1 α 1,β Jn 1(x)=J n − (x) J n − (x), (3.118a) − − α,β 1 α,β 1 α 1,β J (x)= (n+ β)J − (x)+(n+ α)J − (x) . (3.118b) n n+ α + β n n � � α,β a,b n More generally, we can expressJ n in terms of J k k=0, where the expansion coefficients are known as the connection coefficients.{ } Theorem 3.21. Suppose that

n α,β n a,b Jn (x)= ∑ ˆkcJk (x),a,b, α,β > 1. (3.119) k=0 − Then Γ (n+ α +1) (2k+a+b+1) Γ(k+a+b+1) ˆnc = k Γ (n+ α + β +1) Γ (k+a+1) n k m (3.120) − ( 1) Γ (n+k+m+ α + β +1)Γ (m+k+a+1) − . × ∑ m!(n k m)! Γ(k+m+ α +1)Γ (m+2k+a+b+2) m=0 − − Proof. By the Rodrigues’ formula and integration by parts,

1 n 1 α,β a,b a,b ˆkc = a b Jn (x)Jk (x)ω (x)dx , 1 γk �− k 1 ( 1) α,β k a+k,b+k = − a b Jn (x)∂x ω (x) dx 2kk! , 1 γk �− 1 � � 1 k α,β a+k,b+k = a b ∂x Jn (x)ω (x)dx. 2kk! , 1 γk �− 3.2 Jacobi Polynomials 77

Using (3.101) and (3.96) yields

α,β d 1 nc n,k Jα+k,β +k x a+k,b+k x dx ˆk = a,b n k ( )ω ( ) 2kk! 1 − γk �− dα,β Γ (n+ α +1) = n,k k a,b 2 k!γk Γ (n+k+ α + β +1) n k m 1 − ( 1) Γ (n+k+m+ α + β +1) a+m+k,b+k ∑ m− ω dx. × 2 m!(n k m)! Γ(k+m+ α +1) 1 m=0 − − �− a,b α,β Working out γk ,d n,k and the integral respectively by (3.109), (3.102) and (A.6) leads to (3.120). �� α,β α,β Next, we derive some recurrence formulas between J n and ∂xJn . { } { } Theorem 3.22. The Jacobi polynomials satisfy

2 α,β α,β α,β α,β α,β α,β α,β (1 x )∂xJn =A n Jn 1 +B n Jn +Cn Jn+1, (3.121) − − where 2(n+ α)(n+ β)(n+ α + β +1) Aα,β = , (3.122a) n (2n+ α + β)(2n+ α + β +1) 2n(n+ α + β +1) Bα,β = (α β) , (3.122b) n − (2n+ α + β)(2n+ α + β +2) 2n(n+1)(n+ α + β +1) Cα,β = . (3.122c) n − (2n+ α + β +1)(2n+ α + β +2) Proof. This formula follows from (3.100) and (3.116) directly. �� In the Jacobi case, the relation (3.80) takes the following form. Theorem 3.23.

α,β α,β α,β α,β α,β α,β α,β Jn = An ∂xJn 1 + Bn ∂xJn +Cn ∂xJn+1, (3.123) − where � � � 2(n+ α)(n+ β) Aα,β = − , (3.124a) n (n+ α + β)(2n+ α + β)(2n+ α + β +1) 2(α β) � Bα,β = − , (3.124b) n (2n+ α + β)(2n+ α + β +2) 2(n+ α + β +1) Cα�,β = . (3.124c) n (2n+ α + β +1)(2n+ α + β +2) � 78 3 Orthogonal Polynomials and Related Approximation Results

α,β n+1 Proof. We observe from Corollary 3.16 that ∂xJl l=1 forms an orthogonal basis α,β { } ofP n. Hence, we can expressJ n (x)as

n+1 α,β α,β α,β Jn (x)= ∑ el ∂xJl (x), l=1

where 1 α,β 1 α,β 2 α,β α,β el = Jn (x)(1 x )∂xJl (x)ω (x)dx. α,β α,β 1 − γl λl �−

α,β Inserting (3.121) into the above integral and using the orthogonality of J n ,we find that { }

α,β α,β α,β A γn Bn Cα,β =e α,β = n+1 , Bα,β =e α,β = , n n+1 α,β α,β n n α,β γn+1λn+1 λn � � α,β α,β α,β α,β Cn 1γn α,β A =e = − ,e =0,0 l n 2. n n 1 α,β α,β l − ≤ ≤ − γn 1λn 1 − − � Working out the constants yields the coefficients in (3.124). ��

3.2.1.4 Maximum Value

Theorem 3.24. For α,β > 1, set − β α x = − ,q= max( α,β). 0 α + β +1 Then we have max Jα,β 1 n q if q 1 α,β n ( ) , 2 , max Jn (x) = ± ∼ ≥− (3.125) x 1 | | ⎧ α,β 1 1 Jn �(x� ) n 2 �, � if q< , | |≤ ⎨| �� | ∼ − � − 2

where x� is one of the two⎩ maximum points nearest x0. Proof. Define

2 1 2 f (x):= Jα,β (x) + (1 x 2) ∂ Jα,β (x) ,n 1. (3.126) n n α,β x n λn − ≥ � � � � 3.2 Jacobi Polynomials 79

A direct calculation by using (3.90) leads to

2 2 f x α β α β 1 x ∂ Jα,β x n� ( )= α,β ( ) + ( + + ) x n ( ) λn − 2 � �� 2 � α β 1 x x ∂ Jα,β x = α,β ( + + )( 0) x n ( ) . λn − � � Notice that we have the equivalence 1 1 10. − ⇐⇒ 2 2 � �� � We proceed by dividing the parameter range of( α,β) into four different cases. 1 Case I : α,β > . In this case,f (x) 0 (resp.f (x) 0) for allx [ 1,x 0] • − 2 n� ≤ n� ≥ ∈ − (resp.x [x 0,1]). Hence,f n(x) attains its maximum atx= 1, so we have ∈ ±

α,β α,β (3.108) max α,β 1 max Jn (x) =max Jn ( 1) n { }, α,β > . (3.127) x 1 | | ± ∼ − 2 | |≤ (3.106) �� �� � � CaseII: α 1 and 1< β 1 . In this case, the linear function • ≥ − 2 − ≤ − 2 (α β) + (α + β +1)x 0, x [ 1,1], − ≥ ∀ ∈ − which impliesf (x) 0forallx [ 1,1]. Hence, we have n� ≥ ∈ −

α,β α,β α 1 1 max Jn (x) = J n (1) n , α , 1< β . (3.128) x 1 | | | | ∼ ≥ −2 − ≤ −2 | |≤ 1 1 CaseIII: 1< α 2 and β 2 . This situation is opposite to Case II, i.e., • (α β) +− (α + β +≤1) −x 0andf ≥ −(x) 0forallx [ 1,1]. Thus, we have − ≤ n� ≤ ∈ −

α,β α,β β 1 1 max Jn (x) = J n ( 1) n , 1< α , β . (3.129) x 1 | | | − | ∼ − ≤ −2 ≥ −2 | |≤ 1 1 CaseIV: 1< α < and 1< β < . In this case, we have 1

Jacobi polynomials Jacobi polynomials 6 6 4 4 2 2

(x) 0 (x) 0 n n 1,1 1,0 J J −2 −2 −4 −4 −6 −6 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x 1,1 1,0 Fig. 3.1 Jacobi polynomialsJ n (x) (left) andJ n (x) (right) withn=0,1,...,5

3.2.2 Jacobi-Gauss-Type Quadratures

It is straightforward to derive the Jacobi-Gauss-type(i.e., Jacobi-Gauss (JG), Jacobi- Gauss-Radau (JGR) and Jacobi-Gauss-Lobatto (JGL)) integration formulas from the general rules in Sect. 3.1.4. In the Jacobi case, the general quadrature formula (3.33) reads 1 N α,β p(x)ω (x)dx= ∑ p(x j)ω j +E N[p]. (3.130) 1 �− j=0

Recall that if the quadrature errorE N[p]=0, we say ( 3.130)is exact forp.

Theorem 3.25. (Jacobi-Gauss quadrature) The JG quadrature formula( 3.130) is N α,β exact for any p P 2N 1 with the JG nodes x j being the zeros of J (x) and ∈ + j=0 N+1 the corresponding weights given by � � Gα,β ω = N (3.131a) j α,β α,β JN (x j)∂xJN+1(x j) α,β GN = 2 , (3.131b) 2 α,β (1 x ) ∂xJ (x j) − j � N+1 � � where

2α+β (2N+ α + β +2)Γ (N+ α +1)Γ (N+ β +1) Gα,β = , (3.132a) N (N+1)! Γ (N+ α + β +2) 2α+β +1Γ (N+ α +2)Γ (N+ β +2) Gα,β = . (3.132b) N (N+1)! Γ (N+ α + β +2) � 3.2 Jacobi Polynomials 81

Proof. The formula (3.131a) with (3.132a) follows directly from (3.39), and the constant kα,β Gα,β N+1 γα,β N = α,β N kN can be worked out by using (3.95) and (3.109). In order to derive the alternative formula (3.131b) with (3.132b), we first use (3.110) and (3.121) to obtain the recurrence relation

2 α,β (2N+ α + β +2)(1 x )∂xJ (x) − N+1 = (N+1) (2N+ α + β +2)x+ β α Jα,β (x) (3.133) − − N+1 α,β +2(N+ α�+1)(N+ β +1)J N (x). �

α,β Using the factJ N+1(x j) =0, yields

α,β 2N+ α + β +2 2 α,β J (x j) = (1 x )∂xJ (x j). N 2(N+ α +1)(N+ β +1) − j N+1 Plugging it into (3.131a) leads to (3.131b). �� We now consider the Jacobi-Gauss-Radau (JGR) quadrature with the fixed end- pointx 0 = 1. − N Theorem 3.26. (Jacobi-Gauss-Radau quadrature) Let x0 = 1 and x j be − j=1 α,β +1 the zeros of JN (x), and � �

2α+β +1(β +1)Γ 2(β +1)N! Γ (N+ α +1) ω = , (3.134a) 0 Γ (N+ β +2)Γ (N+ α + β +2) α,β +1 1 GN 1 ω j = α β 1 − α β 1 , 1+x j , + , + JN 1 (x j)∂xJN (x j) − (3.134b) α,β +1 1 GN 1 = − ,1 j N. 2 α,β +1 2 (1 x j)(1+x j) [∂xJ (x j)] ≤ ≤ − N� α,β +1 α,β +1 where the constants GN 1 and GN 1 are defined in( 3.132). Then, the quadrature − − formula( 3.130)is exact for any p P 2N. ∈ � Proof. In the Jacobi case, the quadrature polynomialq N defined in (3.48) is orthogonal with respect to the weight function ωα,β +1, so it must be proportional α,β +1 N α,β +1 toJ N . Therefore, the interior nodes x j j=1 are the zeros ofJ N . We now prove (3.134a). The general{ formula} (3.51a) in the Jacobi case reads

1 1 α,β +1 α,β ω0 = JN (x)ω (x)dy. (3.135) Jα,β +1( 1) 1 N − �− 82 3 Orthogonal Polynomials and Related Approximation Results

The formula (3.115) implies

Jα,β +1(x)=a α,β Jα,β (x)+ linear combination of J α,β N , (3.136) N N,0 0 { l }l=1 where � � Γ (α + β +2)Γ (N+ α +1) aα,β = ( 1)N . N,0 − Γ (α +1)Γ (N+ α + β +2) In view ofJ α,β (x) 1, we find from the orthogonality ( 3.88) that 0 ≡

α,β α,β α+β +1 2 aN,0 γ0 2 (β +1)Γ (β +1)N! Γ (N+ α +1) ω0 = = , Jα,β +1( 1) Γ (N+ β +2)Γ (N+ α + β +2) N − where we have worked out the constants by using (3.107) and (3.109). We next prove (3.134b). The Lagrange basis polynomial related tox j is

(1+x)J α,β +1(x) h (x)= N j α,β +1 ∂x (1+x)J (x) (x x j) N x=x j − � α,β +1 �� (1+x)J (x�) = N (3.137) α,β +1 (1+x j)∂xJ (x j)(x x j) N − 1+x = h˜ j(x),1 j N, 1+x j ≤ ≤

˜ N where h j j=1 are the Lagrange basis polynomials associated with the Jacobi-Gauss { } N α,β +1 points x j j=1 (zeros ofJ N ) with the parameters( α,β +1). ReplacingN and β in (3.131a) and (3.132a) byN 1and β +1, yields � � − 1 1 α,β 1 α,β +1 ω j = h j(x)ω (x)dx= h˜ j(x)ω (x)dx 1 1+x j 1 �− �− α,β +1 1 GN 1 = α β 1 − α β 1 (3.138) 1+x j , + , + JN 1 (x j)∂xJN (x j) − α,β +1 1 GN 1 = − ,1 j N. 2 α,β +1 2 (1 x j)(1+x j) [∂xJ (x j)] ≤ ≤ − N� This ends the proof. �� Remark 3.4. A second Jacobi-Gauss-Radau quadrature with a fixed right endpoint xN =1 can be established in a similar manner.

Finally, we consider the Jacobi-Gauss-Lobatto quadrature, which includes two endpointsx= 1 as the nodes. ± 3.2 Jacobi Polynomials 83

Theorem 3.27. (Jacobi-Gauss-Lobatto quadrature) Let x0 = 1,x N =1 and N 1 α,β − − x j j=1 be the zeros of ∂xJN (x), and let � � 2α+β +1(β +1)Γ 2(β +1)Γ (N)Γ (N+ α +1) ω = , (3.139a) 0 Γ (N+ β +1)Γ (N+ α + β +2) 2α+β +1(α +1)Γ 2(α +1)Γ (N)Γ (N+ β +1) ωN = , (3.139b) Γ (N+ α +1)Γ (N+ α + β +2) α+1,β +1 1 GN 2 ω j = − , 1 x 2 α+1,β +1 α+1,β +1 j JN 2 (x j)∂xJN 1 (x j) − − − (3.139c) α+1,β +1 1 GN 2 = − ,1 j N 1, (1 x 2)2 α+1,β +1 2 ≤ ≤ − j ∂xJN 1 (x j) − �− α,β +1� α,β +1 � where the constants GN 1 and GN 1 are defined in( 3.132). Then, the quadrature − − formula( 3.130)is exact for any p P 2N 1. ∈ − The proof is similar to that� of Theorem 3.26 and is left as an exercise (see Problem 3.8). Remark 3.5. The quadrature nodes and weights of these three types of Gaussian α,β α,β N formulas have close relations. Indeed, denote by ξZ,N,j ,ωZ,N,j j=0 with Z=G,R,L the Jacobi-Gauss, Jacobi-Gauss-Radau and Jacobi-Gauss-Lobatto � � quadrature nodes and weights, respectively. Then there hold

α,β +1 α,β α,β +1 α,β ωG,N 1,j 1 ξ = ξ , ω = − − ,1 j N, (3.140) R,N,j G,N 1,j 1 R,N,j α,β +1 − − ≤ ≤ 1+ ξG,N 1,j 1 − − and

α+1,β +1 α,β α+1,β +1 α,β ωG,N 2,j 1 ξ ξ ω − − 1 j N 1 (3.141) L,N,j = G,N 2,j 1 , L,N,j = α+1,β +1 2 , . − − ≤ ≤ − 1 ξG,N 2,j 1 − − − This connection allows us to compute the� interior nodes� and weights of the JGR and JGL quadratures from the JG rule. Moreover, it makes the analysis of JGR and JGL (e.g., the interpolation error) easier by extending the results for JG case.

3.2.3 Computation of Nodes and Weights

Except for the Chebyshev case (see Sect. 3.4), the explicit expressions of the nodes and weights of the general Jacobi-Gauss quadrature are not available, so they have to be computed by numerical means. An efficient algorithm is to use the eigenvalue method described in Theorems 3.4 and 3.6. 84 3 Orthogonal Polynomials and Related Approximation Results

Thanks to the relations (3.140) and (3.141), it suffices to compute the Jacobi- Gauss nodes and weights. Indeed, as a direct consequence of Theorem 3.4, the ze- α,β ros of the Jacobi polynomialJ N+1 are the eigenvalues of the following symmetric tridiagonal matrix

a0 √b1

⎡√b1 a1 √b2 ⎤ ⎢ . . . ⎥ AN+1 = ⎢ ...... ⎥, (3.142) ⎢ ⎥ ⎢ ⎥ ⎢ √bN 1 aN 1 √bN⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ √b a ⎥ ⎢ N N ⎥ ⎣ ⎦ where the entries are derived from (3.25) and the three-term recurrence relation (3.110): β 2 α2 a j = − , (3.143a) (2j+ α + β)(2j+ α + β +2) 4j(j+ α)(j+ β)(j+ α + β) b j = . (3.143b) (2j+ α + β 1)(2j+ α + β)2(2j+ α + β +1) − N Moreover, by Theorem 3.6, the Jacobi-Gauss weights ω j can be obtained by { } j=0 computing the eigenvectors ofA N+1, namely,

α+β +1 α,β 2 2 Γ (α +1)Γ (β +1) 2 ω j = γ Q (x j) = Q (x j) , (3.144) 0 0 Γ (α + β +2) 0 � � � � whereQ 0(x j) is the first component of the orthonormal eigenvector corresponding N to the eigenvaluex j. Notice that weights ω j j=0 may also be computed by using the formula (3.131). { } Alternatively, the zeros of the Jacobi polynomials can be computed by the Newton’s iteration method described in (3.30) and (3.31). The initial approxima- tion can be chosen as some estimates presented below, see, e.g., (3.145). We depict in Fig. 3.2 the distributions of zeros of some sample Jacobi polynomials: 1,1 In(a),thezerosofJ N (x) with variousN • 1 N 1 1,1 In(b),thezeros θ j = cos x j − ofJ (cosθ) with variousN • − j=0 N In(c),thezerosofJ α,α (x) with various α • � 15 � In(d),thezerosofJ α,0(x) with various α • 15 We observe from (a) and (b) in Fig. 3.2 that the zeros x j (arranged in descending order) of the Jacobi polynomials are nonuniformly{ } distributed in 1 ( 1,1), while θ j = cos− x j are nearly equidistantly located in(0, π). More pre- − { } 2 cisely, the nodes (inx) cluster near the endpoints with spacing density likeO(N − ), 1 and are considerably sparser in the inner part with spacingO(N − ). This feature is 3.2 Jacobi Polynomials 85

quantitatively characterized by Theorem 8.9.1 of Szego¨ (1975), which states that for α,β > 1, −

a node distribution in (−1,1) b node distribution in (0,π)

−1 −0.5 0 0.5 1 0 1 2 3 x θ

c node distribution (α=β) d node distribution (β=0) 2 2

1.5 1.5 α 1 α 1

0.5 0.5

0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x Fig. 3.2 Distributions of Jacobi-Gauss quadrature nodes

1 1 cos− x j = θ j = (j+1) π +O(1) ,j=0,1,...,N, (3.145) N+1 � � whereO(1) is uniformly bounded for all valuesj=0,1,...,N, andN=1,2,3,.... We see that near the endpointsx= 1 (i.e., θ =0, π), ±

2 2 2 1 x = sin θ j =O(N − ),j=0,N. − j

2 Hence, the node spacing in the neighborhood ofx= 1 behaves likeO(N − ). In particular, for the case ± 1 1 1 1 α , β , (3.146) −2 ≤ ≤ 2 − 2 ≤ ≤ 2 86 3 Orthogonal Polynomials and Related Approximation Results

Theorem 6.21.2 of Szego¨ (1975) provides the bounds

2j+1 2j+2 θ j ,0 j N, (3.147) 2N+3 ≤ ≤ 2N+3 ≤ ≤

where the equality holds only when α = β = 1 or α = β = 1 . − − 2 − 2 For a fixedj, we can viewx j =x j(N; α,β) as a function ofN, α and β, and observe from (c) and (d) in Fig. 3.2 that for a givenN, the nodes exhibit a tendency to move towards the center of the interval as α and/or β increases. This is predicted by Theorem 6.21.1 of Szego¨ (1975):

∂x j ∂x j <0, >0,0 j N. (3.148) ∂α ∂β ≤ ≤

In particular, if α = β,

∂x j <0,j=0,1,...,[N/2]. (3.149) ∂α

3.2.4 Interpolation and Discrete Jacobi Transforms

N Let x j,ω j j=0 be a set of Jacobi-Gauss-type nodes and weights. As in Sect. 3.1.5, we{ can define} the corresponding interpolation operator, discrete inner product and α,β discrete norm, denoted byI N , , N,ωα,β and N,ωα,β , respectively. The exactness of the quadratures�· ·� implies �·�

u,v α,β = (u,v) α,β , u v P , (3.150) � � N,ω ω ∀ · ∈ 2N+δ where δ =1,0, 1 for JG, JGR and JGL, respectively. Accordingly, we have −

u α,β = u α,β , u P N, forJGandJGR. (3.151) � �N,ω � � ω ∀ ∈ Although the above identity does not hold for the JGL case, we have the following equivalence. Lemma 3.3.

α + β +1 u ωα,β u N,ωα,β 2+ u ωα,β , u P N. (3.152) � � ≤� � ≤ � N � � ∀ ∈

Proof. For anyu P N, we write ∈ N 1 u(x)= ˆuJα,β (x), with ˆu= u,J α,β . ∑ l l l α,β l ωα,β l=0 γl � � 3.2 Jacobi Polynomials 87

By the orthogonality of the Jacobi polynomials and the exactness (3.150),

N u 2 ˆu2 α,β ωα,β = ∑ l γl , � � l=0 (3.153) N 1 u 2 − ˆu2 α,β ˆu2 Jα,β J α,β N,ωα,β = ∑ l γl + N N , N N,ωα,β . � � l=0 � � To estimate the last term, we define

2 α,β 2 1 2 α,β ψ(x)=[J (x)] + (1 x ) ∂xJ (x) . N N2 − N � � 2N One verifies readily that ψ P 2N 1, since the leading termx cancels out. There- 2 ∈ α,β− fore, using the fact(1 x )∂xJ (x j) =0, and the exactness ( 3.150), we derive − j N α,β α,β α,β α,β J ,J = 1, ψ α,β = (1,ψ) α,β = J ,J N N N,ωα,β � �N,ω ω N N ωα,β

� � �α,β � 1 α,β α,β (3.97) λN α,β + ∂xJN , ∂xJN = 1+ γN . N2 ωα+1,β+1 N2 � � � � Hence, by (3.91),

α + β +1 Jα,β ,J α,β = 2+ γα,β . (3.154) N N N,ωα,β N N � � � � Inserting it into (3.153) leads to the desired result. �� We now turn to the discrete Jacobi transforms. Since the interpolationpolynomial α,β IN u P N, we write ∈ N α,β α,β α,β IN u (x)= ∑ ˜un Jn (x), (3.155) n=0 � � α,β N where the coefficients ˜un n=0 are determined by the forward discrete Jacobi transform. { } Theorem 3.28.

1 N ˜uα,β = u(x )Jα,β (x )ω , (3.156) n α,β ∑ j n j j δn j=0

α,β α,β where δn = γn for0 n N 1, and ≤ ≤ − γα,β , forJGandJGR, α,β N δN = ⎧ 2+ α+β +1 γα,β , for JGL. ⎨ N N � � ⎩ 88 3 Orthogonal Polynomials and Related Approximation Results

Proof. This formula follows directly from Theorem 3.9 and (3.154). �� By takingx=x j in (3.155), the backward discrete Jacobi transformis carriedout by

N α,β α,β α,β u(x j) = (IN u)(x j) = ∑ ˜un Jn (x j),0 j N. (3.157) n=0 ≤ ≤ In general, the discrete transforms (3.156)-(3.157)can be performedby a matrix– vector multiplication routine in aboutN 2 flops. Some techniques to reduce the com- putational complexity toN(logN) α (with some positive α) are suggested in Potts et al. (1998), Tygert (2010).

3.2.5 Differentiation in the Physical Space

N N Let x j be a set of Jacobi-Gauss-type points, and let h j be the associ- { } j=0 { } j=0 ated Lagrange basis polynomials. Suppose thatu P N is an approximation to the underlying solution, and we write ∈

N u(x)= ∑ u(x j)h j(x). j=0

As shown in Sect. 3.1.6, the differentiation ofu can be done through a matrix–vector multiplication: u(m) =D mu,m 1, (3.158) ≥ (k) (k) (k) (k) T (0) whereu = (u (x0),u (x1),...,u (xN )) ,u=u , and the first-order differ- entiation matrix:

D= dk j =h �j(xk) k,j=0,1,...,N . Hence, it suffices to compute the� entries of the� first-order differentiation matrixD, whose explicit formulas can be derived from Theorem 3.11.

3.2.5.1 Jacobi-Gauss-Lobatto Differentiation Matrix

In this case, the quadrature polynomial defined in (3.74) reads

2 α+1,β +1 Q(x)=(1 x )JN 1 (x). − − To simplify the notation, we write

α+1,β +1 J(x):= ∂xJN 1 (x). (3.159) − 3.2 Jacobi Polynomials 89

One verifies readily that (note:x 0 = 1 andx N = 1): − 2( 1)N 1Γ (N+ β +1) − − ,j=0, Γ (N)Γ (β +2) ⎧ 1 x 2 J x 1 j N 1 Q�(x j) = ⎪( j) ( j), , ⎪ − ≤ ≤ − ⎨⎪ 2Γ (N+ α +1) − ,j=N. Γ (N)Γ (α +2) ⎪ ⎪ DifferentiatingQ(x) yields⎩⎪

α+1,β +1 α+1,β +1 2 2 α+1,β +1 Q��(x)= 2J N 1 (x) 4x ∂xJN 1 (x)+(1 x )∂x JN 1 (x) − − − − − − (3.90) α+1,β +1 α+1,β +1 = [(α β) + (α + β)x]J(x) ( λN 1 +2)J N 1 (x). − − − − α+1,β +1 N 1 Recalling that J N 1 (x j) =0 j=−1 , and using the formulas (3.94), (3.107) and (3.100) to work{ out− the constants,} we find

2 α N(N+ α + β +1) Γ (N+ β +1) − ,j=0, ( 1)N+1Γ (N)Γ (β +3) ⎧ � − � α β + (α + β)x J(x ),1 j N 1, Q��(x j) = ⎪ j j ⎪ − ≤ ≤ − ⎨⎪�2 β N(N+ α + β�+1) Γ (N+ α +1) − ,j=N. Γ (N)Γ (α +3) ⎪ � � ⎪ Applying the⎩⎪ general formulas in Theorem 3.11, the entries of the first-order JGL differentiation matrixD are expressed as follows. (a). The first column (j= 0):

α N(N+ α + β +1) − ,k=0, 2(β +2) ⎧ N 1 ( 1) − Γ (N)Γ (β +2) dk0 = ⎪ − (1 x k)J(xk),1 k N 1, (3.160) ⎪ 2Γ (N+ β +1) − ≤ ≤ − ⎪ ⎨( 1)N Γ (β +2)Γ (N+ α +1) − ,k=N. ⎪ 2 Γ (α +2)Γ (N+ β +1) ⎪ ⎪ (b). The second⎩⎪ to theN-th column (1 j N 1): ≤ ≤ − 2( 1)NΓ (N+β+1) − 2 ,k=0, Γ (N)Γ (β+2)(1 x j)(1+x j) J(x j) ⎧ 2 − (1 x )J(xk) 1 ⎪ − k ,k=j,1 k N 1, ⎪ 2 ⎪(1 x j )J(x j) xk x j � ≤ ≤ − dk j=⎪ − − (3.161) ⎪α β+(α+β)xk ⎪ − ,1 k=j N 1, ⎨ 2(1 x2) ≤ ≤ − − k ⎪ 2Γ (N+α+1) ⎪ − 2 ,k=N. ⎪Γ (N)Γ (α+2)(1 x j) (1+x j)J(x j) ⎪ − ⎪ ⎩⎪ 90 3 Orthogonal Polynomials and Related Approximation Results

(c). The last column (j=N):

( 1)N+1 Γ (α +2)Γ (N+ β +1) − ,k=0, 2 Γ (β +2)Γ (N+ α +1) ⎧ ⎪ ⎪Γ (N)Γ (α +2) dkN = ⎪ (1+x k)J(xk),1 k N 1, (3.162) ⎪2Γ (N+ α +1) ≤ ≤ − ⎨⎪ N(N+ α + β +1) β ⎪ − ,k=N. ⎪ 2(α +2) ⎪ ⎪ ⎩⎪

3.2.5.2 Jacobi-Gauss-Radau Differentiation Matrix

α,β +1 In this case, the quadrature polynomial in (3.74) isQ(x)=(1+x)J N (x). Denot- α,β +1 ingJ(x)= ∂xJN (x), one verifies that

( 1)NΓ (N+ β +2) − ,j=0, N!Γ (β +2) Q�(x j) = ⎧ ⎪ ⎨(1+x j)J(x j),1 j N. ≤ ≤ ⎪ We obtain from (3.100) and⎩⎪ (3.107) that

N 1 α,β +1 ( 1) − (N+ α + β +2)Γ (N+ β +2) Q��(x0) =2 ∂xJ ( 1)= − . N − Γ (N)Γ (β +3) Moreover, by (3.90),

α,β +1 2 α,β +1 α,β +1 Q��(x)=2 ∂xJN (x)+(1+x) ∂x JN (x)=2 ∂xJN (x)

1 α,β +1 α,β +1 α,β +1 + α β 1+( α + β +3)x ∂xJ (x) λ J (x) . 1 x − − N − N N − �� � � α,β +1 N In view of JN (x j) =0 j=1, we derive that � 1 � Q��(x j) = α β +1+( α + β +1)x j J(x j),1 j N. 1 x j − ≤ ≤ − � � 3.2 Jacobi Polynomials 91

Applying the general results in Theorem 3.11 leads to

N(N+ α + β +2) − ,k=j=0, 2(β +2) ⎧ ⎪ ⎪ N!Γ (β +2) ⎪ J(xk),1 k N,j=0, ⎪( 1)NΓ (N+ β +2) ≤ ≤ ⎪ − ⎪ ⎪( 1)N+1Γ (N+ β +2) 1 ⎪ k 0 1 j N dk j = ⎪ − 2 , = , , (3.163) ⎪ N!Γ (β +2) (1+x j) J(x j) ≤ ≤ ⎨⎪ (1+x )J(x ) 1 ⎪ k k ,1 k=j N, ⎪(1+x )J(x ) x x ⎪ j j k j ≤ � ≤ ⎪ − ⎪ ⎪α β +1+( α + β +1)x k ⎪ − ,1 k=j N. ⎪ 2(1 x 2) ≤ ≤ ⎪ − k ⎪ ⎩⎪ 3.2.5.3 Jacobi-Gauss Differentiation Matrix

α,β In this case, the quadrature polynomial in (3.74) isQ(x)=J N+1(x). One verifies by using (3.90) that

2 α,β 1 α,β ∂x JN+1(x j) = 2 α β + (α + β +2)x j ∂xJN+1(x j),0 j N. 1 x j − ≤ ≤ − � � Once again, we derive from Theorem 3.11 that

α,β ∂xJ (xk) 1 N+1 0 k j N α,β , = , ⎧ J x xk x j ≤ � ≤ ∂x N+1( j) − dk j = ⎪ (3.164) ⎪ ⎨⎪ α β + (α + β +2)x k − 2 ,1 k=j N. 2(1 x k) ≤ ≤ ⎪ − ⎪ As a numerical illustration,⎩⎪ we consider the approximation of the derivatives of u(x)= sin(4πx),x [ 1,1] by the Jacobi-Gauss-Lobatto interpolation associated N ∈ − with x j with α = β =1. More precisely, let { } j=0 N 1,1 u(x) u N(x)=I N u(x)= ∑ u(x j)h j(x) P N. (3.165) ≈ j=0 ∈

In Fig. 3.3a, we plotu � (solid line) versusu N� (x) (“ ”) andu �� (solid line) versus N u (x)(“�”) at x j withN=38. In Fig. 3.3b, we· depict the errors log u N�� { } j=0 10 � � − u 1,1 (“ ”) and log u u 1,1 (“ ”) against variousN We observe N� N,ω 10 �� N�� N,ω . � that� the errors◦ decay exponentially.� − � � � � � 92 3 Orthogonal Polynomials and Related Approximation Results a b u(k) vs. u(k) with k=1,2 Convergence of differentiation N 200 0 −2 100 −4 −6 0

(error) −8 10 −10

−100 log −12 −200 −14 −1 −0.5 0 0.5 1 20 25 30 35 40 X N Fig. 3.3 Convergence of Jacobi differentiation in the physical space

3.2.6 Differentiation in the Frequency Space

We now describe the spectral differentiation by manipulating the expansion coeffi- cients as in Sect. 3.1.7. For anyu P N, we write ∈ N N 1 α,β − (1) α,β u(x)= ∑ ˆunJn (x) P N,u �(x)= ∑ ˆun Jn (x) P N 1. n=0 ∈ n=0 ∈ −

(1) The process of differentiation in the frequency space is to express ˆun in terms { } of ˆun . {Thanks} to the recurrence formula (3.123), the corresponding coefficients in the relation (3.80) are α,β ˜ α,β α,β ˜an = An , bn = Bn ,˜c n = Cn , where Aα,β Bα,β and Cα,β are given in (3.124a)–(3.124c), respectively. n , n n � � � (1) N Hence, by Theorem 3.12, the coefficients ˆun n=0 can be exactly evaluated by the backward� � recurrence� formula { } ˆu ˆu(1) =0,ˆu (1) = N , N N 1 α,β − CN 1 ⎧ − ⎪ (1) 1 (1) α,β (1) (3.166) ⎪ ˆu = ˆu Bα,β ˆu A ˆu , ⎪ n 1 α,β n �n n n+1 n+1 ⎪ − − − ⎨ Cn 1 − � � ⎪ n=N 1,N� 2,...,2,1.� ⎪ � − − ⎪ N In summary, given⎩⎪ the physical values u(x j) j=0 at a set of Jacobi-Gauss-type N { N } points x j j=0, the evaluation of u �(x j) j=0 can be carried out in the following three steps:{ } { } 3.3 Legendre Polynomials 93

N Find the coefficients ˆun n=0 by using the forward discrete Jacobi transform • (3.156). { } (1) N 1 Compute the coefficients ˆun n=−0 by using (3.166). • { } N Find the derivative values u �(x j) j=0 by using the backward discrete Jacobi • transform( 3.157). { } Higher-order derivatives can be computed by repeating the above procedure.

a u(k) vs. u(k) with k=1,2 b Convergence of differentiation N 0 1 −2 0 −4

−1 −6 (error) 8

10 − −2

log −10 −3 −12 −4 −14 −1 −0.5 0 0.5 1 5 10 15 20 25 30 35 40 45 50 X N Fig. 3.4 Convergence of Jacobi differentiation in the frequency space

As an illustrative example, we fix the Jacobi index to be(1,1), consideru(x)= 1/(1+2x 2),x [ 1,1], and approximate its derivatives by taking the derivatives, in the frequency∈ − space, of its interpolation polynomial: N 1,1 1,1 u(x) u N(x)=I N u(x)= ∑ ˜unJn (x) P N. (3.167) ≈ n=0 ∈ We observe from Fig. 3.4 that the errors decay exponentially, similar to the differ- entiation in the physical space as shown in Fig. 3.3.

3.3 Legendre Polynomials

We discuss in this section an important special case of the Jacobi polynomials– then Legendre polynomials 0,0 Ln(x)=J (x),n 0,x I=( 1,1). n ≥ ∈ − The distinct feature of the Legendre polynomials is that they are mutually orthog- onal with respect to the uniform weight function ω(x) 1. The first six Legendre polynomials and their derivatives are plotted in Fig. 3.5≡. Since most of them can be derived directly from the corresponding properties of the Jacobi polynomials by taking α = β =0, we merely collect some relevant formulas without proof. 94 3 Orthogonal Polynomials and Related Approximation Results

Three-term recurrence relation: •

(n+1)L n+1(x)=(2n+1)xL n(x) nL n 1(x),n 1, (3.168) − − ≥ and the first few Legendre polynomials are

L0(x)=1,L 1(x)=x,

1 2 1 3 L2(x)= (3x 1),L 3(x)= (5x 3x). 2 − 2 − The Legendre polynomial has the expansion • [n/2] 1 l (2n 2l)! n 2l Ln(x)= ( 1) − x − , (3.169) 2n ∑ − 2nl!(n l)!(n 2l)! l=0 − − and the leading coefficient is

(2n)! kn = . (3.170) 2n(n!)2

Sturm-Liouville problem: • 2 (1 x )L� (x) � + λnLn(x)=0, λn =n(n+1). (3.171) − n Equivalently, � �

2 (1 x )L��(x) 2xL � (x)+n(n+1)L n(x)=0. (3.172) − n − n Rodrigues’ formula: • n 1 d 2 n Ln(x)= (x 1) ,n 0. (3.173) 2nn! dxn − ≥ � � Orthogonality: • 1 2 Ln(x)Lm(x)dx= γnδmn, γn = , (3.174a) 1 2n+1 �− 1 2 Ln� (x)Lm� (x)(1 x )dx= γnλnδmn. (3.174b) 1 − �− Symmetric property: • n n Ln( x)=( 1) Ln(x),L n( 1)=( 1) . (3.175) − − ± ±

Hence,L n(x) is an odd (resp. even) function, ifn is odd (resp. even). Moreover, we have the uniform bound

Ln(x) 1, x [ 1,1],n 0. | | ≤ ∀ ∈ − ≥ 3.3 Legendre Polynomials 95

Derivative recurrence relations: •

(2n+1)L n(x)=L n� +1(x) L n� 1(x),n 1, (3.176a) − − ≥ n 1 − Ln� (x)= ∑ (2k+1)L k(x), (3.176b) k=0 k+n odd n 2 − 1 Ln��(x)= ∑ k+ n(n+1) k(k+1) Lk(x), (3.176c) k=0 2 − k+n even� �� � 2 n(n+1) (1 x )Ln� (x)= Ln 1(x) L n+1(x) . (3.176d) − 2n+1 − − � � The boundary values of the derivatives: • 1 n 1 L� ( 1)= ( 1) − n(n+1), (3.177a) n ± 2 ± n L��( 1)=( 1) (n 1)n(n+1)(n+2)/8. (3.177b) n ± ± −

Legendre polynomials First−order derivative 1 15

10 0.5 5 (x) (x)

0 n n ′ L L 0 −0.5 −5

−1 −10 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x x Fig. 3.5 The first six Legendre polynomials and their first-order derivatives

3.3.1 Legendre-Gauss-Type Quadratures

The Legendre-Gauss-type quadrature formulas can be derived from the Jacobi ones in the previous section. 96 3 Orthogonal Polynomials and Related Approximation Results

N Theorem 3.29. Let x j,ω j be a set of Legendre-Gauss-typenodes and weights. { } j=0 For the Legendre-Gauss (LG) quadrature, • N x j j=0 are the zeros of LN+1(x); 2 (3.178) � � 0 j N ω j = 2 2 , . (1 x )[L (x j)] ≤ ≤ − j N� +1 For the Legendre-Gauss-Radau (LGR) quadrature, • N x j j=0 are the zeros of LN (x)+L N+1(x); (3.179) � � 1 1 x j ω j = 2 − 2 ,0 j N. (N+1) [LN (x j)] ≤ ≤

For the Legendre-Gauss-Lobatto (LGL) quadrature, • N 2 x j are the zeros of(1 x )L� (x); j=0 − N (3.180) � � 2 1 ω j = 2 ,0 j N. N(N+1) [LN (x j)] ≤ ≤ With the above quadrature nodes and weights, there holds

1 N p(x)dx= ∑ p(x j)ω j, p P 2N+δ , (3.181) 1 ∀ ∈ �− j=0

where δ =1,0, 1 for LG, LGR and LGL, respectively. − Proof. Therule(3.181) with (3.178) follows directly from Theorem 3.25 with α = β =0. We now prove (3.179). The formula (3.116b) implies

0,1 (1+x)J N (x)=L N (x)+L N+1(x). (3.182)

N Hence, we infer from Theorem 3.26 that the nodes x j j=0 are the zeros ofL N (x)+ L (x), and the formulas of the weights are N+1 � �

2 1 1 x 0 ω0 = 2 = 2 − 2 , (N+1) (N+1) [LN (x0)]

2(2N+1) 1 ω j = ,1 j N. N(N+1) 0,1 0,1 ≤ ≤ (1+x j) JN 1(x)∂xJN (x) − x=x j � �� � � 3.3 Legendre Polynomials 97

0,1 To derive the equivalent expression in (3.179), we deduce from the fact J N (x j) = 0 N that { } j=1 0,1 0,1 (3.133) 2N(N+1) JN 1 (x j) ∂xJ (x j) = − N 2N+1 1 x 2 − j

(3.182) 2N(N+1) LN 1(x j) +L N(x j) = − 2 , 2N+1 (1+x j)(1 x ) − j which, together with (3.182), leads to

2 0,1 0,1 2N(N+1) LN 1(x j) +L N(x j) 1 (1+x j)JN 1(x j)∂xJN (x j) = − . − 2N+1 1+x j 1 x j � � −

Due toL N (x j) +L N+1(x j) =0for1 j N, using the three-term recurrence rela- tion (3.168) gives ≤ ≤

2N+1 N+1 LN 1(x j) = x jLN (x j) LN+1(x j) − N − N 2N+1 N+1 = x L (x ) + L (x ) N j N j N N j 2N+1 = (1+x j)LN (x j) L N(x j). N − A combination of the above facts leads to (3.179). We now turn to the derivation of (3.180). By Theorem 3.27 with α = β =0, 2 ω = ωN = , 0 N(N+1) (3.183) 8 1 ω j = ,1 j N 1. N+1 2 1,1 1,1 ≤ ≤ − (1 x j)JN 2(x j)∂xJN 1(x j) − − − 1,1 N In view of J N 1(x j) =0 j=1, we derive from (3.133) that { − } 2 1,1 1,1 (1 x j)∂xJN 1(x j) =NJ N 2(x j),1 j N 1. − − − ≤ ≤ − As a consequence of (3.98) and the above equality, we find that

2 1,1 1,1 1,1 2 4 2 (1 x j)JN 2(x j)∂xJN 1(x j) =N JN 2(x j) = LN� 1 (x j) . − − − − N − � � � � 98 3 Orthogonal Polynomials and Related Approximation Results

N Differentiating (3.168) and using (3.176a) and the fact L (x j) =0 , yields { N� } j=1 2N+1 N+1 LN� 1(x j) = LN (x j) LN� +1(x j) − N − N 2N+1 N+1 = LN (x j) LN� 1(x j) + (2N+1)L N(x j) N − N − N+�1 � = (2N+1)L N(x j) LN� 1(x j), − − N − which leads to

LN� 1 (x j) = NL N(x j),1 j N 1. − − ≤ ≤ − Consequently,

2 1,1 1,1 2 (1 x j)JN 2 (x j)∂xJN 1(x j) =4NL N (x j). − − − Plugging it into the second formula of (3.183) gives the desired result. ��

3.3.2 Computation of Nodes and Weights

As a special case of (3.142), the interior Legendre-Gauss-type nodes are the eigen- values of the following Jacobian matrix:

a0 √b1

⎡√b1 a1 √b2 ⎤ ⎢ . . . ⎥ AM+1 = ⎢ ...... ⎥, (3.184) ⎢ ⎥ ⎢ ⎥ ⎢ √bM 1 aM 1 √bM⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ √b a ⎥ ⎢ M M ⎥ where ⎣ ⎦ j2 ForLG:a j =0,b j = ,M=N. • 4j 2 1 1 − j(j+1) ForLGR:a j = ,b j = ,M=N 1. • (2j+1)(2j+3) (2j+1) 2 − j(j+2) ForLGL:a j =0,b j = ,M=N 2. • (2j+1)(2j+3) − The quadrature weights can be evaluated by using the formulas in Theorem 3.29. Alternatively, as a consequence of (3.144), the quadrature weights can be computed from the first component of the orthonormal eigenvectors ofA M+1. The eigenvalue method is well-suited for the Gauss-quadratures of low or mod- erate order. However, for high-order quadratures, the eigenvalue method may suffer 3.3 Legendre Polynomials 99

from round-off errors, so it is advisable to use a root-finding iterative approach. To fix the idea, we restrict our attention to the commonly used Legendre-Gauss-Lobatto case and compute the zeros ofL N� (x). In this case, the Newton method( 3.31) reads

k LN� (x j) xk+1 =x k ,k 0, j j L xk ⎧ − N�� ( j) ≥ (3.185) ⎨⎪givenx 0,1 j N 1. j ≤ ≤ − ⎩⎪ To avoid evaluating the values ofL N�� ,weuse( 3.171) to derive that

L (x) (1 x 2)L (x) N� = − N� . L (x) 2xL (x) N(N+1)L N(x) N�� N� − For an iterative method, it is essential to start with a good initial approximation. In Lether (1978), an approximation of the zeros ofL N (x) is given by

N 1 1 28 5 σk = 1 − 3 4 39 2 cosθk +O(N − ), (3.186) − 8N − 384N − sin θk � � �� where 4k 1 θk = − π,1 k N. 4N+2 ≤ ≤ Notice from Corollary 3.4 (the interlacing property) that there exists exactly one zero ofL N� (x) between two consecutive zeros ofL N (x). Therefore, we can take the initial guess as

σ j + σ j 1 x0 = + ,1 j N 1. (3.187) j 2 ≤ ≤ −

N+1 We pointout thatduetoL N� ( x)=( 1) LN� (x), the computationalcostof (3.185) can be halved. − − N After finding the nodes x j j=0, we can compute the corresponding weights by the formula (3.180): { } 2 1 w(x)= 2 . (3.188) N(N+1) LN (x)

It is clear thatw �(x j) =0 for 1 j N 1. In other words, the interior nodes are the extremes ofw(x). We plot≤ ≤ the− graph ofw(x) withN= 8 in Fig. 3.6a. As a consequence, for a small perturbation of the nodes, we can obtain very accurate values ω j =w(x j) even for very largeN. In Fig. 3.6b, we depict the locations of the Legendre-Gauss-Lobatto nodes 8 8 x j , and θ j = arccosxj . We see that θ j distribute nearly equidistantly { } j=0 { } j=0 { } 100 3 Orthogonal Polynomials and Related Approximation Results along the upper half unit circle (i.e., in[0, π]). The projection of θ j onto[ 1,1] { } − 2 yields the clustering of points x j near the endpointsx= 1 with spacingO(N ). { } ± −

a Behavior of w(x) b Distribution of nodes (N=8) 6 θ θ 4 θ5 4 1 3 θ θ2 6 2 w(x) θ1 0 0.5 θ7 (x) & w(x) 8

′ −2 L ′ 0 L 8(x) −4 x0 x1 x2 x3 x4 x5 x6 x7 x8 −6 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1 x

Fig. 3.6(a) Behavior ofw(x)in( 3.188) withN= 8; (b) Distribution of the Legendre-Gauss- Lobatto nodes withN=8

We tabulate in Table 3.2 some samples of the LGL nodes and weights with N=8,16 (note thatx N j = x j and ωN j = ω j) computed by the aforementioned method. − − −

Table 3.2 LGL nodes and weights Nodesx j Weights ω j 1.000000000000000e+00 2.777777777777778e-02 8.997579954114601e-01 1.654953615608056e-01 6.771862795107377e-01 2.745387125001617e-01 3.631174638261782e-01 3.464285109730462e-01 0.000000000000000e+00 3.715192743764172e-01 1.000000000000000e+00 7.352941176470588e-03 9.731321766314184e-01 4.492194054325414e-02 9.108799959155736e-01 7.919827050368709e-02 8.156962512217703e-01 1.105929090070281e-01 6.910289806276847e-01 1.379877462019266e-01 5.413853993301015e-01 1.603946619976215e-01 3.721744335654770e-01 1.770042535156577e-01 1.895119735183174e-01 1.872163396776192e-01 0.000000000000000e+00 1.906618747534694e-01

3.3.3 Interpolation and Discrete Legendre Transforms

N Given a set of Legendre-Gauss-type quadrature nodes and weights x j,ω j , we { } j=0 define the associated interpolation operatorI N , discrete inner product , N and �· ·� discrete norm N, as in Sect. 3.1.5. �·� 3.3 Legendre Polynomials 101

Thanks to the exactness of the Legendre-Gauss-type quadrature (cf. (3.181)), we have u,v N = (u,v), u v P , (3.189) � � ∀ · ∈ 2N+δ where δ =1,0, 1 for LG, LGR and LGL, respectively. Consequently, −

u N = u , u P N forLGandLGR. (3.190) � � � � ∀ ∈ Although the above formula does not hold for LGL, we derive from Lemma 3.3 with α = β = 0 the following equivalence:

1 u u N 2+N u , u P N . (3.191) � �≤� � ≤ − � � ∀ ∈ Moreover, as a direct consequence of� (3.154), we have 2 L ,L = . (3.192) N N N N � � We now turn to the discrete Legendre transforms. The Lagrange interpolation polynomialI Nu P N, so we write ∈ N (IN u)(x)= ∑ ˜unLn(x), n=0

where the (discrete) Legendre coefficients ˜un are determined by the forward dis- crete Legendre transform: { }

N 1 u,L n N ˜un = ∑ u(x j)Ln(x j)ω j = � �2 ,0 n N, (3.193) γn Ln ≤ ≤ j=0 � �N 2 2 where γn = for 0 n N, except for LGL case, γN = . On the other hand, 2n+1 ≤ ≤ N given the expansion coefficients ˜un , the physical values u(x j) can be computed by the backward discrete Legendre{ transform} : { }

N u(x j) = (IN u)(x j) = ∑ ˜unLn(x j),0 j N. (3.194) n=0 ≤ ≤

Assuming that Ln(x j) j,n=0,1,...,N have been precomputed, the discrete Legendre transforms (3.194) and (3.193) can be carried out by a standard matrix–vector mul- � � tiplication routine in aboutN 2 flops. The cost of the discrete Legendre transforms n can be halved, due to the symmetry:L n(x j) = ( 1) Ln(xN j ). To illustrate the convergence of Legendre interpolation− − approximations, we con- sider the test function:u(x)= sin(k πx). Writing

∞ sin(kπx)= ∑ ˆunLn(x),x [ 1,1], (3.195) n=0 ∈ − 102 3 Orthogonal Polynomials and Related Approximation Results

we can derive from the property of the Bessel functions (cf. Watson (1966)) that

1 ˆun = (2n+1)J n+1/2(kπ)sin(nπ/2),n 0, (3.196) √2k ≥ whereJ ( ) is the Bessel function of the first kind. Using the asymptotic formula n+1/2 · 1 ex ν Jν (x) , ν 1, ν R, (3.197) ∼ 2πν 2ν � ∈ � � we find that the exponential decay of the expansion coefficients occurs when the mode ekπ 1 n> . (3.198) 2 − 2

N We now approximateu byI N u= ∑n=0 ˜unLn(x), and consider the error in the coefficients ˆun ˜un . We observe from Fig. 3.7a that the errors between the exact and discrete| expansion− coefficients decay exponentially whenN>ek π/2, and it � verifies the estimate�

max ˆun ˜un ˆuN+1 forN 1. (3.199) 0 n N − ∼ � ≤ ≤ � � � � In Fig. 3.7b, we depict the exact expansion coefficients ˆun (marked by “ ”) and ◦ the discrete expansion coefficients ˜un (marked by “�”) against the subscriptn, and in Fig. 3.7c, we plot the exact solution versus its interpolation. Observe thatI Nu provides an accurate approximation tou as long asN>ek π/2. As with the Fourier case, when a discontinuous function is expanded in Legen- dre series, the Gibbs phenomena occur in the neighborhood of a discontinuity. For example, the Legendre series expansion of the sign function sgn(x) is

∞ ( 1)n(4n+3)(2n)! x L x sgn( )= ∑ − 2n+1 2n+1( ). (3.200) n=0 2 (n+1)!n!

One verifies readily that the expansion coefficients behave like

(4n+3)(2n)! 1 ˆu2n 1 = ,n 1. (3.201) | + | 22n+1(n+1)!n! � √n �

In Fig. 3.7d, we plot the numerical approximationI N u(x) and sgn(x) in the interval [ 0.3,0.3], which indicates a Gibbs phenomenon nearx=0. − 3.3 Legendre Polynomials 103

a Errors between coefficients b Coefficients 4 100 2 10−4 k=32 0 10−8 Errors −2 10−12 k=16 10−16 −4 0 1 2 0 40 80 120 N/(kπ) n

I u(x) versus u(x) c N d INu(x) versus u(x) 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 x x

Fig. 3.7(a) Error max 0 n N ˆun ˜un &ˆuN+1 (solid line) vs.N/(k π) withk= 16,24,32; (b) ˆu n ≤ ≤ | − | vs. ˜un withk= 32 andN= 128; (c)u(x) vs.I N u(x),x [ 0.3,0.3] withk= 32 andN= 128; ∈ − (d)u(x) = sgn(x) vs.I N u(x),x [ 0.3,0.3] withk= 32 andN= 64 ∈ −

3.3.4 Differentiation in the Physical Space

N Givenu P N and its values at a set of Legendre-Gauss-type points x j j=0, let N ∈ { } h j j=0 be the associated Lagrange basis polynomials. According to the general approach{ } described in Sect. 3.1.6, we have

u(m) =D mu,m 1, (3.202) ≥ where

(m) (m) (m) T (0) D= dk j =h �j(xk) 0 k,j N ,u = u (x0),...,u (xN ) ,u=u . ≤ ≤ We derive� below explicit� representations� of the entries ofD for the� three different cases by using the general formulas of the Jacobi polynomials in Sect. 3.2.5.

For the Legendre-Gauss-Lobattocase (x0 = 1 andx N = 1): The general formu- • las (3.160)–(3.162) for JGL in Sect. 3.2.5 with− α = β = 0 lead to the reduced 104 3 Orthogonal Polynomials and Related Approximation Results

2 1,1 1,1 formulas involving(1 x )∂xJN 1 and(1 x) ∂xJN 1. Using (3.100) and (3.172) leads to − − ± −

2 1,1 2 2 2 (1 x j)∂xJN 1(x j) = (1 x j)∂x LN (x j) − − N+1 − = 2NL N(x j),1 j N 1, − ≤ ≤ − and

1,1 LN (x j) (1 x j)∂xJN 1(x j) = 2N ,1 j N 1. ± − 1 x j ≤ ≤ − − ∓ Plugging the above in (3.160)–(3.162) with α = β = 0, we derive

N(N+1) ,k=j=0, − 4 ⎧ ⎪LN (xk) 1 ⎪ ,k=j,0 k,j N, d = ⎪LN (x j) xk x j � ≤ ≤ (3.203) k j ⎪ − ⎪0,1 k=j N 1, ⎨ ≤ ≤ − N(N+1) ⎪ k=j=N. ⎪ 4 ⎪ ⎪ ⎩⎪ For the Legendre-Gauss-Radau case (x0 = 1): The general formula ( 3.163) in • the case of α = β = 0 can be simplified to −

N(N+2) ,k=j=0, − 4 ⎧ ⎪ xk (N+1)L N(xk) dk j = ⎪ 2 + 2 ,1 k=j N, (3.204) ⎪1 x (1 x )Q (xk) ≤ ≤ ⎪ − k − k � ⎨Q (x ) 1 � k ,k=j, ⎪Q (x j) xk x j � ⎪ � − ⎪ ⎪ ⎩ 0,1 whereQ(x) =L N (x)+L N+1 (x) (which is proportional to(1+x)J N (x)). For k=j, we derive from Theorem 3.11 that

Q��(xk) dkk = ,0 k N. 2Q�(xk) ≤ ≤ To avoid computing the second-order derivatives, we obtain from (3.172) that

2xkQ�(xk) +2(N+1)L N(xk) Q��(xk) = ,1 k N. 1 x 2 ≤ ≤ − k Fork=j=0, we can work out the constants by using ( 3.177). 3.3 Legendre Polynomials 105

For the Legendre-Gauss case: The general formula (3.164) in the case of α = • β = 0 reduces to

L� (xk) 1 N+1 ,k=j, L (x ) x x � d = N� +1 j k j (3.205) k j ⎧ x − ⎪ k ,k=j. ⎨ 1 x 2 − k ⎪ In all cases, the differentiation⎩ matrixD is a full matrix, soO(N 2) flops are N N (N+1) needed to compute u (x j) from u(x j) . Also note that sinceu (x) 0 � j=0 j=0 ≡ N+1 R N+1 for anyu P N, we� haveD � u=0 for� anyu� . Hence, the only eigenvalue ofD is zero∈ which has a multiplicityN+1. ∈

3.3.5 Differentiation in the Frequency Space

Givenu P N, we write ∈ N u(x)= ∑ ˆukLk(x) P N, k=0 ∈ and N N (1) (1) u�(x)= ∑ ˆukLk� (x)= ∑ ˆuk Lk(x) with ˆuN =0. k=1 k=0 Thanks to (3.176a), we find

N N 1 (1) (1) − (1) 1 u� = ∑ ˆuk Lk =ˆu0 + ∑ ˆuk (Lk� +1 L k� 1) k=0 k=1 2k+1 − − (1) N 1 (1) (1) ˆuN 1 − ˆuk 1 ˆuk+1 = − L� + − L� . 2N 1 N ∑ 2k 1 − 2k+3 k − k=1 � − � Since L are orthogonal polynomials (cf. (3.174b)), comparing the coefficients of { k� } Lk� leads to the backward recursive relation:

(1) (1) ˆuk+1 ˆuk 1 = (2k 1) ˆuk + ,k=N 1,N 2,...,1, − − 2k+3 − − (3.206) (1) (1) � � ˆuN =0,ˆu N 1 = (2N 1)ˆuN. − − Higher-order differentiations can be performed by the above formula recursively. 106 3 Orthogonal Polynomials and Related Approximation Results 3.4 Chebyshev Polynomials

In this section, we consider another important special case of the Jacobi polynomi- als – Chebyshev polynomials (of the first kind), which are proportional to Jacobi 1/2, 1/2 polynomials J n− − and are orthogonal with respect to the weight function { 2 1/2 } ω(x)=(1 x )− . The three-term− recurrence relation for the Chebyshev polynomials reads:

Tn+1(x)=2xT n(x) T n 1(x),n 1, (3.207) − − ≥ withT 0(x)=1andT 1(x)=x. The Chebyshev polynomials are eigenfunctions of the Sturm-Liouville problem:

2 2 2 1 x 1 x T �(x) � +n Tn(x)=0, (3.208) − − n or equivalently, � �� � 2 2 (1 x )T ��(x) xT �(x)+n Tn(x)=0. (3.209) − n − n While we can derive the properties of Chebyshev polynomials from the gen- eral properties of Jacobi polynomials with( α,β) = ( 1/2, 1/2), it is more con- venient to explore the relation between Chebyshev polynomials− − and trigonometric functions. Indeed, using the trigonometric relation

cos((n+1) θ) +cos((n 1) θ) =2cos θ cos(nθ), − and taking θ = arccosx, we find that cos(narccosx) satisfies the three-term recur- rence relation (3.207), and it is 1,x forn=0,1, respectively. Thus, by an induction argument, cos(narccosx) is also a polynomial of degreen with the leading coeffi- n 1 cient 2 − (Fig. 3.8). We infer from Theorem 3.1 of the uniqueness that

Tn(x)= cosnθ, θ = arccosx,n 0,x I. (3.210) ≥ ∈ This explicit representation enables us to derive many useful properties. An immediate consequence is the recurrence relation 1 1 2Tn(x)= Tn�+1(x) Tn� 1(x),n 2. (3.211) n+1 − n 1 − ≥ − One can also derive from (3.210) that

n n Tn( x)=( 1) T(x),T n( 1)=( 1) , (3.212a) − − ± ± 2 Tn(x) 1, T �(x) n , (3.212b) | | ≤ | n | ≤ 2 n n (1 x )Tn�(x)= Tn 1(x) Tn+1(x), (3.212c) − 2 − − 2 2Tm(x)Tn(x)=T m+n(x)+Tm n(x),m n 0, (3.212d) − ≥ ≥ 3.4 Chebyshev Polynomials 107 and

n 1 2 T �( 1)=( 1) − n , (3.213a) n ± ± 1 n 2 2 T ��( 1)= ( 1) n (n 1). (3.213b) n ± 3 ± − It is also easy to show that

1 1 cnπ Tn(x)Tm(x) dx= δmn, (3.214) 1 √1 x 2 2 �− − wherec 0 =2andc n =1forn 1. Hence, we find from ( 3.208) that ≥ 1 2 2 n cnπ Tn�(x)Tm� (x) 1 x dx= δmn, (3.215) 1 − 2 �− � 2 i.e., T n�(x) are mutually orthogonal with respect to the weight function √1 x . We{ can} obtain from (3.211) that −

n 1 − 1 Tn�(x)=2n ∑ Tk(x), (3.216a) k=0 ck k+nodd n 2 − 1 2 2 Tn��(x)= ∑ n(n k )Tk(x). (3.216b) k=0 ck − k+n even

cos(12θ) 1

0.5

0

−0.5

−1 T12 (x+0.5)

0 0.5 1 1.5 2 2.5 3

Fig. 3.8 Left: curves ofT 12(x+1.5) and cos(12 θ); Right: we plotT n(x) radially, increase the radius for each value ofn, and fill in the areas between the curves (Trott (1999), pp. 10 and 84)

Another remarkable consequence of (3.210) is that the Gauss-type quadrature nodes and weights can be derived explicitly. 108 3 Orthogonal Polynomials and Related Approximation Results

N Theorem 3.30. Let x j,ω j j=0 be a set of Chebyshev-Gauss-type quadrature nodes and weights. { } For Chebyshev-Gauss (CG) quadrature, • (2j+1) π π x j = cos , ω j = ,0 j N. − 2N+2 N+1 ≤ ≤ For Chebyshev-Gauss-Radau (CGR) quadrature, • 2π j x j = cos ,0 j N, − 2N+1 ≤ ≤ π 2π ω0 = , ω j = ,1 j N. 2N+1 2N+1 ≤ ≤ For Chebyshev-Gauss-Lobatto (CGL) quadrature, • π j π x j = cos , ω j = ,0 j N. − N ˜cjN ≤ ≤

where˜c0 =˜cN =2 and˜c j =1for j=1,2,...,N 1. − With the above choices, there holds

1 1 N p(x) dx= ∑ p(x j)ω j, p P 2N+δ , (3.217) 1 √1 x 2 ∀ ∈ �− − j=0 where δ =1,0, 1 for the CG, CGR and CGL, respectively. − In the Chebyshev case, the nodes θ j = arccos(xj) are equally distributed on { } [0,π], whereas x j are clustered in the neighborhood ofx= 1 with density 2 { } ± O(N− ), for instance, for the CGL points

2 π 2 π π 1 x 1 =1 cos = 2sin forN 1. − − N 2N � 2N2 � For more properties of Chebyshev polynomials, we refer to Rivlin (1974).

3.4.1 Interpolation and Discrete Chebyshev Transforms

N Given a set of Chebyshev-Gauss-typequadrature nodes and weights x j,ω j , we { } j=0 define the associated interpolation operatorI N , discrete inner product , N,ω and �· ·� discrete norm N,ω, as in Sect. 3.1.5. Thanks to the�·� exactness of the Chebyshev-Gauss-type quadrature, we have

u,v N,ω = (u,v) ω , uv P , (3.218) � � ∀ ∈ 2N+δ 3.4 Chebyshev Polynomials 109 where δ =1,0, 1 for CG, CGR and CGL, respectively. Consequently, −

u N,ω = u ω, u P N, forCGandCGR. (3.219) � � � � ∀ ∈ Although the above identity does not hold for the CGL, the following equivalence follows from Lemma 3.3:

u ω u N,ω √2 u ω, u P N. (3.220) � � ≤� � ≤ � � ∀ ∈ Moreover, a direct computation leads to

π N cos2 jπ TN ,TN N,ω = ∑ = π. (3.221) � � N j=0 ˜cj We now turn to the discrete Chebyshev transforms. To fix the idea, we only consider the Chebyshev-Gauss-Lobatto case. As a special family of Jacobi poly- nomials, the transforms can be performed via a matrix–vector multiplication with O(N2) operations as usual. However, thanks to (3.210), they can be carried out with O(N log2 N) operations via FFT. Givenu C[ 1,1], letI Nu be its Lagrange interpolation polynomial relative to the CGL points,∈ − and we write

N (IN u)(x)= ∑ ˜unTn(x) P N, n=0 ∈ where ˜u n are determined by the forward discrete Chebyshev transform (cf. Theorem{ 3.9} ):

2 N 1 n jπ ˜un = ∑ u(x j)cos ,0 n N. (3.222) ˜ncN j=0 ˜cj N ≤ ≤

On the other hand, given the expansion coefficients ˜u n , the physical values { } u(x j) are evaluated by the backward discrete Chebyshev transform: { } N N n jπ u(x j) = (IN u)(x j) = ∑ ˜unTn(x j) = ∑ ˜un cos ,0 j N. (3.223) n=0 n=0 N ≤ ≤ Hence, it is clear that both the forward transform (3.222) and backward transform (3.223) can be computed by using FFT inO(N log 2 N) operations. Let us conclude this part with a discussion of point-per-wavelength required for the approximation using Chebyshev polynomials. We have

∞ sin(kπx)= ∑ ˆunTn(x),x [ 1,1], (3.224) n=0 ∈ − with 2 ˆun :=ˆun(k)= Jn(kπ)sin(nπ/2),n 0, (3.225) cn ≥ 110 3 Orthogonal Polynomials and Related Approximation Results whereJ n( ) is again the Bessel functionof the first kind.Hence, using the asymptotic formula (3.197· ), we find that the exponential decay of the expansion coefficients occurs when ekπ n> , (3.226) 2 which is similar to (3.198) for the Legendre expansion.

3.4.2 Differentiation in the Physical Space

Givenu P N and its values at a set of Chebyshev-Gauss-type collocation points N ∈ N x j j=0, let h j(x) j=0 be the associated Lagrange basis polynomials. According to{ the} general{ results} stated in Sect. 3.1.6, we have

u(m) =D mu,m 1, (3.227) ≥ where

(m) (m) (m) T (0) D= dk j =h �j(xk) 0 k,j N ,u = u (x0),...,u (xN ) ,u=u . ≤ ≤ The entries� of the first-order� differentiation� matrixD can be determined� by the ex- plicit formulas below.

For the Chebyshev-Gauss-Lobatto case (x0 = 1 andx N = 1): • − 2N2 +1 ,k=j=0, ⎧− 6 ⎪ ˜c ( 1)k+j ⎪ k ⎪ − ,k=j,0 k,j N, d = ⎪ ˜cj xk x j � ≤ ≤ (3.228) k j ⎪ x− ⎪ k ⎨ 2 ,1 k=j N 1, −2(1 x k) ≤ ≤ − 2 − ⎪2N +1 ⎪ ,k=j=N, ⎪ 6 ⎪ ⎪ where ˜0c =˜cN =2and ˜⎩⎪cj =1for1 j N 1. ≤ ≤ − For the Chebyshev-Gauss-Radau case (x0 = 1): • − N(N+1) ,k=j=0, − 3 ⎧ x (2N+1)T (x ) ⎪ k + N k ,1 k=j N, dk j = ⎪ 2 2 (3.229) ⎪2(1 x k) 2(1 x k)Q�(xk) ≤ ≤ ⎪ − − ⎨Q (x ) 1 � k ,k=j, ⎪Q�(x j) xk x j � ⎪ − ⎪ ⎩⎪ 3.4 Chebyshev Polynomials 111

whereQ(x)=T N (x)+TN+1 (x). To derive (3.229), we find from Theorem 3.26 N 1/2,1/2 that x j are the zeros of(1+x)J − (x). In view of the correspondence: { } j=0 N

1/2, 1/2 1/2, 1/2 JN− − (x)=J N− − (1)TN (x), (3.230)

one verifies by using (3.116b) that

1/2,1/2 1/2, 1/2 (1+x)J N− (x)=J N− − (1) TN (x)+TN+1(x) . � � Hence, fork=j, we find from Theorem 3.11 that

Q��(xk) dkk = ,0 k N. 2Q�(xk) ≤ ≤

To avoid evaluating the second-order derivatives, we derive from (3.209) and the factQ(x k) =0 that

xkQ�(xk) + (2N+1)T N(xk) Q��(xk) = ,1 k N. 1 x 2 ≤ ≤ − k Hence, xk (2N+1)T N(xk) dkk = 2 + 2 ,1 k N. 2(1 x ) 2(1 x )Q (xk) ≤ ≤ − k − k �

The formula for the entryd 00 follows directly from the Jacobi-Gauss-Radau case with α = β =0. For the Chebyshev-Gauss case: •

T � (xk) 1 N+1 ,k=j, T (x ) x x � d = N�+1 j k j (3.231) k j ⎧ x − ⎪ k ,k=j. ⎨ 2(1 x 2) − k ⎪ ⎩⎪ 3.4.3 Differentiation in the Frequency Space

Now, we describe the FFT algorithm for Chebyshev spectral differentiation. Let us start with the conventional approach. Given

N u(x)= ∑ ˆukTk(x) P N, (3.232) k=0 ∈ 112 3 Orthogonal Polynomials and Related Approximation Results

we derive from (3.211) that

N N (1) (1) u� = ∑ ˆukTk� = ∑ ˆuk Tk (with ˆuN =0) k=1 k=0 N 1 (1) (1) − (1) Tk�+1 Tk� 1 =ˆu0 +ˆu1 T1 + ∑ ˆuk − (3.233) k 2 2(k+1) − 2(k 1) = � − � (1) N 1 ˆuN 1 − 1 (1) (1) = − TN� + ∑ ck 1 ˆuk 1 ˆuk+1 Tk�, 2N k=1 2k − − − � � wherec 0 =2andc k =1fork 1. Since T are mutually orthogonal, we compare ≥ { k�} the expansion coefficients in terms of T and find that ˆu(1) can be computed { k�} { k } from ˆuk via the backward recurrence relation: { } (1) (1) ˆuN =0,ˆu N 1 =2NˆuN, − (3.234) (1) (1) ˆuk 1 = 2kˆuk +ˆuk+1 /ck 1,k=N 1,...,1. − − − Higher-order derivatives can� be evaluated� recursively by this relation. N N Notice that given u(x j) j=0 at the Chebyshev-Gauss-Lobatto points x j j=0, { }N { } the computation of u (x j) through the process of differentiation in the phys- { � } j=0 ical space requiresO(N 2) operations due to the fact that the differentiation matrix (see the previous section) is full. However, thanks to the fast discrete Chebyshev transforms between the physical values and expansion coefficients, one can com- N N pute u (x j) from u(x j) inO(N log N) operations as follows: { � } j=0 { } j=0 2 Compute the discrete Chebyshev coefficients ˆuk from u(x j) using (3.222) in • { } { } O(N log2 N) operations. (1) Compute the Chebyshev coefficients ˆuk ofu � using (3.234) inO(N) opera- • tions. { } (1) (1) Compute u (x j) from ˆu using (3.223) (with ˆu ,u (x j) in place of • { � } { k } { k � } ˆuk,u(x j) ) inO(N log N) operations. { } 2 To summarize, thanks to its relation with Fourier series (cf. (3.210)), the Chebyshev polynomials enjoy several distinct advantages over other Jacobi polyno- mials: The nodes and weights of Gauss-type quadratures are given explicitly, avoiding • the potential loss of accuracy at largeN when computing them through a numer- ical procedure. The discrete Chebyshev transforms can be carried out using FFT inO(N log 2 N) • operations. Thanks to the fast discrete transforms, the derivatives as well as nonlinear terms • can also be evaluated inO(N log 2 N) operations. However, the fact that the Chebyshev polynomials are mutually orthogonal with respect to a weighted inner product may induce complications in analysis and/or implementations of a Chebyshev spectral method. 3.5 Error Estimates for Polynomial Approximations 113 3.5 Error Estimates for Polynomial Approximations

The aim of this section is to perform error analysis, in anisotropic Jacobi-weighted Sobolev spaces, for approximating functions by Jacobi polynomials. These results play a fundamental role in analysis of spectral methods for PDEs. More specifically, we shall consider: Inverse inequalities for Jacobi polynomials • Estimates for the best approximation by series of Jacobi polynomials • Error analysis of Jacobi-Gauss-type polynomial interpolations • Many results presented in this section with estimates in anisotropic Jacobi- weighted Sobolev spaces are mainly based on the papers by Guo and Wang (2001, 2004) (also see Funaro (1992)). Similar estimates in standard Sobolev spaces can be found in the books by Bernardi and Maday (1992a, 1997) and Canuto et al. (2006).

3.5.1 Inverse Inequalities for Jacobi Polynomials

Since all norms of a function in any finite dimensional space are equivalent, we have

∂xφ C N φ , φ PN, � �≤ � � ∀ ∈ which is an example of inverse inequalities. The inverse inequalities are very useful for analyzing spectral approximations of nonlinear problems. In this context, an important issue is to derive the optimal constantC N . Recall that the notationA�B means that there exists a generic positive constantc, independent ofN and any function, such thatA cB. The first inverse≤ inequality relates two norms weighted with different Jacobi weight functions.

Theorem 3.31. For α,β > 1 and any φ P N, we have − ∈ α,β ∂xφ α+1,β+1 λ φ α,β , (3.235) � �ω ≤ N � �ω � and m m ∂ φ α+m,β+m �N φ α,β ,m 1, (3.236) � x �ω � �ω ≥ α,β where λN =N(N+ α + β +1).

Proof. For any φ PN, we write ∈ N 1 ˆα,β α,β ˆα,β 1 α,β α,β φ(x)= ∑ φn Jn (x) with φn = φJn ω dx. (3.237) α,β 1 n=0 γn �− 114 3 Orthogonal Polynomials and Related Approximation Results

Hence, by the orthogonality of Jacobi polynomials,

N 2 α,β ˆα,β 2 φ ωα,β = ∑ γn φn . � � n=0 | | Differentiating (3.237) and using the orthogonality (3.97), we obtain

N 2 α,β α,β α,β 2 φ � = λ γ φˆ � �ωα+1,β+1 ∑ n n | n | n=1 (3.238) N α,β α,β ˆα,β 2 α,β 2 λN ∑ γn φn λN φ ωα,β , ≤ n=1 | | ≤ � � which yields (3.235). Using the above inequality recursively leads to

m 1 1/2 m − α+k,β +k ∂x φ ωα+m,β+m ∏ λN k φ ωα,β . (3.239) � � ≤ k=0 − � � � � Hence, we obtain (3.236) by using (3.91). �� If the polynomial φ vanishes at the endpointsx= 1, i.e., ± 0 φ P := u P N :u( 1)=0 , (3.240) ∈ N ∈ ± the following inverse inequality holds.� � Theorem 3.32. For α,β > 1 and any φ P 0, − ∈ N

∂xφ α,β �N φ α 1,β 1 . (3.241) � �ω � �ω − − Proof. We refer to Bernardi and Maday (1992b) for the proof of α = β, and Guo and Wang (2004) for the derivation of the general case. Here, we merely sketch the 2 proof of α = β =0. Since φ/(1 x ) P N 2, we write − ∈ − N 1 2 − ˜ φ(x)/(1 x ) = ∑ φnLn� (x). − n=1 Thus, by (3.174b), N 1 2 − 2 φ 1, 1 = n(n+1) γn φ˜n , ω− − ∑ � � n=1 | |

where γn =2/(2n+1).Inviewof( 3.171), we have

N 1 N 1 − ˜ 2 − ˜ φ �(x)= ∑ φn (1 x )Ln� (x) � = ∑ n(n+1) φnLn(x), n=1 − − n=1 � � 3.5 Error Estimates for Polynomial Approximations 115

and by (3.174a), N 1 2 − 2 2 ˜ 2 ∂xφ = ∑ n (n+1) γn φn . � � n=1 | | Thus, we have 2 2 ∂xφ N(N 1) φ 1, 1 . (3.242) � � ≤ − � �ω− − This gives (3.241) with α = β =0. �� The inverse inequality (3.236) is an algebraic analogy to the trigonometric in- verse inequality (2.44), and both of them involve “optimal” constantC N =O(N). However, the norms in (3.236) are weighted with different weight functions. In most applications, we need to use inverse inequalities involving the same weighted norms. For this purpose, we present an inverse inequality with respect to the Legen- dre weight function ω(x) 1 (cf. Canuto and Quarteroni (1982)). ≡ Theorem 3.33. For any φ P N, ∈ 1 ∂xφ (N+1)(N+2) φ . (3.243) � �≤ 2 � � Proof. Using integration by parts, (3.174a), (3.175) and (3.177a), we obtain 1 1 2 1 (3.244) Ln� (x) dx=L n(x)Ln� (x) 1 Ln��(x)Ln(x)dx=n(n+1). 1 − − 1 �− �− � � � Hence, by (3.174a), �

n(n+1)(2n+1) 3/2 Ln� = Ln (n+1) Ln ,n 0. (3.245) � � � 2 � �≤ � � ≥

Next, for any φ PN, we write ∈ N 1 ˆ ˆ 1 φ(x)= ∑ φnLn(x) with φn = n+ φ(x)Ln(x)dx, n=0 2 1 � ��− so we have N 2 2 ˆ 2 φ = ∑ φn . � � n=0 2n+1 | | On the other hand, we obtain from (3.244) and the Cauchy–Schwarz inequality that

N N ˆ ˆ ∂xφ ∑ φn Ln� ∑ φn n(n+1) � �≤ n=0 | |� �≤ n=0 | | � N 1/2 N 1/2 2 ˆ 2 ∑ φn ∑ n(n+1) n+1/2 ≤ n=0 2n+1 | | n=0 � � � � �� N 1/2 (N+1)(N+2) ∑ (n+1) 3 φ φ . ≤ n=0 � �≤ 2 � � � � This ends the proof. �� 116 3 Orthogonal Polynomials and Related Approximation Results

Remark 3.6. The factor N2 in( 3.243) is sharp in the sense that for any positive integer N, there exists a polynomial ψ P N and a positive constant c independent of N such that ∈ 2 ∂xψ cN ψ . (3.246) � �≥ � �

Indeed, taking ψ(x)=L N� (x), one verifies readily by using integration by parts, (3.174a),( 3.177),( 3.100),( 3.94) and( 3.107) that

1 2 1 LN�� (x) dx= LN�� (x)LN� (x) L N���(x)LN (x) 1 − 1 �− � ��− (3.247) � � 1 � 2 = (N 1)N(N+1)(N+2)(N � +N+3), 12 −

which, together with( 3.244), implies

1 2 L�� = (N 1)(N+2)(N +N+3) L � . � N� 2√3 − � N� � This justifies the claim. We now consider the extension of (3.243) to the Jacobi polynomials. We ob- serve from the proof of Theorem 3.33 that the use of (3.244) allows for a sim- 1 α,β 2 α,β ple derivation of (3.243). However, the explicit formula for 1 ∂xJn ω dx for general( α,β) is much more involved, although one can derive− them by using � (3.119)–(3.120) (and (3.216a) for the Chebyshev case). We refer to� Guo (1998a)� for the following result, and leave the proof of the Chebyshev case as an exercise (see Problem 3.21).

Theorem 3.34. For α,β > 1 and any φ P N, − ∈ 2 ∂xφ α,β �N φ α,β . � �ω � �ω

3.5.2 Orthogonal Projections

A common procedure in error analysis is to compare the numerical solutionu N with a suitable orthogonal projection πN u (or interpolationI Nu) of the exact solutionu in some appropriate Sobolev space with the norm S (cf. Remark 1.7), and use the triangle inequality, �·�

u u N S u πNu S + πNu u N S. � − � ≤� − � � − �

Hence, one needs to estimate the errors u πNu S and I Nu u S. Such estimates involving Jacobi polynomials will be the� main− concern� of� this− section.� 3.5 Error Estimates for Polynomial Approximations 117

LetI=( 1,1), and let ωα,β (x)=(1 x) α (1+x) β with α,β > 1, be the Jacobi weight function− as before. For anyu L− 2 (I), we write − ∈ ωα,β

∞ α,β u,Jn α,β u(x)= ˆuα,β Jα,β (x) with ˆuα,β = ω , (3.248) ∑ n n n α,β n=0 � γn �

α,β α,β 2 where γn = J n . � �ωα,β 2 α,β 2 Define theL -orthogonal projection π :L (I) P N such that ωα,β N ωα,β → α,β π u u,v =0, v P N, (3.249) N − ωα,β ∀ ∈ or equivalently, � � N α,β α,β α,β (πN u)(x)= ∑ ˆun Jn (x). (3.250) n=0 α,β We find from Theorem 3.14 that πN u is the best polynomial approximation ofu in L2 (I). ωα,β α,β To measure the truncation error πN u u, we introduce the non-uniformly (or anisotropic) Jacobi-weighted Sobolev space:−

Bm (I):= u: ∂ ku L 2 (I),0 k m ,m N, (3.251) α,β x ∈ ωα+k,β+k ≤ ≤ ∈ � � equipped with the inner product, norm and semi-norm

m k k u,v Bm = ∑ ∂x u,∂x v ωα+k,β+k , α,β k=0 (3.252) � � �1/2 � m m m u B = u,u Bm , u B = ∂x u ωα+m,β+m . � � α,β α,β | | α,β � �

m � � The spaceB α,β (I) distinguishes itself from the usual weighted Sobolev space Hm (I) (cf. Appendix B) by involving different weight functions for derivatives of ωα,β different orders. It is obvious thatH m (I) is a subspace ofB m (I), that is, for any ωα,β α,β m 0and α,β > 1, ≥ − u Bm c u Hm . � � α,β ≤ � � ωα,β Before presenting the main result, we first derive from (3.101) to (3.102) and the orthogonality (3.109) that

1 k α,β k α,β α+k,β +k α,β ∂x Jn (x)∂x Jl (x)ω (x)dx=h n,k δnl, (3.253) 1 �− 118 3 Orthogonal Polynomials and Related Approximation Results

where forn k, ≥ α,β α,β 2 α+k,β +k hn,k = (dn,k ) γn k − 2α+β +1Γ (n+ α +1)Γ (n+ β +1)Γ (n+k+ α + β +1) (3.254) = . (2n+ α + β +1)(n k)! Γ 2(n+ α + β +1) − Summing (3.253) for all 0 k m, we find that the Jacobi polynomials are orthog- ≤m≤ onal in the Sobolev spaceB α,β (I), namely,

α,β α,β Jn ,J =0, ifn=l. (3.255) l Bm α,β � � � Now, we are ready to state the first fundamental result. Theorem 3.35. Let α,β > 1. For anyu B m (I), − ∈ α,β if0 l m N+1, we have • ≤ ≤ ≤ ∂ l(πα,β u u) x N − ωα+l,β+l � � (3.256) � � (N m+1)! (l m)/2 m c − (N+m) − ∂x u α+m,β+m , ≤ � (N l+1)! ω − � � if m>N+1, we have � � • ∂ l(πα,β u u) x N − ωα+l,β+l (3.257) � � e/2 N l+1 � c(2�πN) 1/4 − ∂ N+1 u , − N x ωα+N+1,β+N+1 ≤ � � � � � where c 1forN 1. � � ≈ � Proof. Denote ˜m= min m,N+1 . Thanks to the orthogonality (3.253)–(3.254), { } ∞ ku 2 hα,β ˆuα,β 2 k 0 (3.258) ∂x ωα+k,β+k = ∑ n,k n , , � � n=k | | ≥ so we have ∞ l α,β u u 2 hα,β ˆuα,β 2 ∂x(πN ) ωα+l,β+l = ∑ n,l n � − � n=N+1 | | hα,β ∞ n,l α,β α,β 2 max ∑ hn,˜m ˆun (3.259) ≤ n N+1 ⎧hα,β ⎫ | | ≥ ⎨ n,˜m⎬ n=N+1 α,β hN+1,l⎩ ⎭2 ∂ ˜mu . ≤ α,β x ωα+˜mβ, +˜m hN+1,˜m � � � � 3.5 Error Estimates for Polynomial Approximations 119

By (3.254),

α,β hN+1,l Γ (N+l+ α + β +2)(N ˜m+1)! = − . (3.260) hα,β Γ (N+˜m+ α + β +2)(N l+1)! N+1,˜m − Using the Stirling’s formula (A.7) yields

Γ (N+l+ α + β +2) 1 (N+˜m)l ˜m. (3.261) ∼= ˜m l∼= − Γ (N+˜m+ α + β +2) (N+˜m+ α + β +2) − Correspondingly, hα,β N+1,l 2 (N ˜m+1)! l ˜m c − (N+˜m) − , (3.262) hα,β ≤ (N l+1)! N+1,˜m − wherec 1. A combination of the above estimates leads to ≈ l α,β 2 2 (N ˜m+1)! l ˜m ˜m 2 ∂ (π u u) c − (N+˜m) − ∂ u . (3.263) � x N − � ωα+l,β+l ≤ (N l+1)! x ωα+˜mβ, +˜m − � � Finally, if 0 l m N+1, then ˜m=m, so ( 3.256) follows.� � On the other hand, ifm>N+1, then≤ ≤ ˜m≤=N+1, and the estimate ( 3.257) follows from (3.263) and Stirling’s formula (A.8). �� Remark 3.7. In contrast with error estimates for finite elements or finite differences, the convergence rate of spectral approximations is only limited by the regularity of the underlying function. Therefore, we made a special effort to characterize the explicit dependence of the errors on the regularity index m. For any fixed m, the estimate( 3.256) becomes

l α,β l m m ∂ (π u u) l l �N − ∂ u m m , (3.264) x N − ωα+ ,β+ x ωα+ ,β+ � � � � which is the typical� convergence� rate found in the� literature.� Hereafter, the factor

(N m+1)! − ,0 m N+1, � N! ≤ ≤ frequently appears in the characterization of the approximation errors. For a quick reference,

(1 m)/2 (N m+1)! 1 N − − = ≤ N! N(N 1)...(N (m 2)) (3.265) � − − − (1 m)/2 (N m+2) − ,� ≤ − 120 3 Orthogonal Polynomials and Related Approximation Results

so for m=o(N)(in particular, for fixed m), we have

(N m+1)! (1 m)/2 − ∼= N − . (3.266) � N! Some other remarks are also in order. α,β Theorem 3.35 indicates that the truncated Jacobi series πN u is the best poly- • nomial approximation ofu in bothL 2 (I) and the anisotropic Jacobi-weighted ωα,β l Sobolev spaceB α,β (I). α,β It must be pointed out that the truncation error πN u u measured in the usual • weighted Sobolev spaceH l (I) (withl 1) does not− have an optimal order ωα,β of convergence. Indeed, one can always≥ find a function such that its truncated Jacobi series converges inL 2 (I), but diverges inH 1 (I). For instance, we ωα,β ωα,β 0,0 takeu=L N+1 L N 1 , and notice that πN u= L N 1 and ∂xu=(2N+1)L N.It is clear that − − − −

0,0 (3.244) √N ∂x(π u u) = L � = (N+1)(N+2) ∂xu . � N − � � N+1� ≥ 2 � � � In general, we have the following estimates: for α > 1 and 0 l m, − ≤ ≤ α,α 2l m 1/2 m π u u �N − − ∂ u α+m,α+m . (3.267) N − l,ωα,α � x �ω � � This estimate for� the Legendre� and Chebyshev cases was derived in Canuto and Quarteroni (1982), and in Guo (2000) for the general case with α,β > 1. − SinceH l (I) is a Hilbert space, the best approximation polynomial foru is the ωα,β orthogonal projection ofu uponP N under the inner product

l k k u,v l,ωα,β = ∑ ∂x u,∂x v ωα,β , (3.268) k=0 � � � � which induces the norm ofH l (I). In fact, this type of approximation l,ωα,β ωα,β results are often needed� in·� analysis of spectral methods for second-order elliptic PDEs. Therefore, we consider below theH 1 -orthogonal projection. Denote the ωα,β inner product inH 1 (I)by ωα,β

1 a (u,v):= u�,v � + u,v , u,v H (I), α,β ωα,β ωα,β ∀ ∈ ωα,β � � 1 � � 1 and define the orthogonal projection π :H (I) P N by N,α,β ωα,β → 1 a (π u u,v)=0, v P N. (3.269) α,β N,α,β − ∀ ∈ 3.5 Error Estimates for Polynomial Approximations 121

1 By definition, πN,α,β u is the best approximation ofu in the sense that 1 πN,α,β u u 1,ωα,β = inf φ u 1,ωα,β . (3.270) � − � φ PN � − � ∈ By using the fundamental Theorem 3.35, we can derive the following estimate.

m 1 Theorem 3.36. Let α,β > 1.If ∂xu B − (I), then for1 m N+1, − ∈ α,β ≤ ≤

1 (N m+1)! (1 m)/2 m π u u α,β c − (N+m) − ∂x u α+m 1,β+m 1 , � N,α,β − � 1,ω ≤ N! � �ω − − � (3.271) where c is a positive constant independent of m,N andu.

Proof. Let πα,β be theL 2 -orthogonal projection uponP as defined in N 1 ωα,β N 1 (3.249). Set − − x α,β φ(x)= πN 1u�(y)dy+ ξ , (3.272) 1 − �− where the constant ξ is chosen such that φ(0)=u(0).Inviewof( 3.270), we derive from the inequality (B.43) and Theorem 3.35 that

1 π u u α,β φ u α,β c ( φ u) � α,β � N,α,β − � 1,ω ≤� − � 1,ω ≤ � − �ω α,β (N m+1)! (1 m)/2 m c πN 1u� u � ωα,β c − (N+m) − ∂x u ωα+m 1,β+m 1 . ≤ � − − � ≤ � N! � � − − This completes the proof. �� While the estimate (3.271) is optimal in theH 1 -norm, it does not imply an ωα,β optimal order in theL 2 -norm. An optimal estimate in theL 2 -norm can be ωα,β ωα,β obtained by using a duality argument, which is also known as the Aubin-Nitsche technique (see, e.g., Ciarlet (1978)). The first step is to show the regularity of the solution for an auxiliary problem. Lemma 3.4. Let α,β > 1. For each g L 2 (I), there exists a unique ψ − ∈ ωα,β ∈ H1 (I) such that ωα,β 1 a (ψ,v)=(g,v) α,β , v H (I). (3.273) α,β ω ∀ ∈ ωα,β Moreover, the solution ψ H 2 (I) and satisfies ∈ ωα,β

ψ α,β � g α,β . (3.274) � �2,ω � � ω Proof. Thebilinearforma ( , ) is the inner productof the HilbertspaceH 1 (I), α,β ωα,β so the existence and uniqueness· · of the solution ψ of (3.273) follows from the Riesz representation theorem (see Appendix B).Takingv= ψ in (3.273) and using the Cauchy–Schwarz inequality leads to

ψ α,β � g α,β . (3.275) � �1,ω � � ω 122 3 Orthogonal Polynomials and Related Approximation Results

By takingv D(I) in ( 3.273) (whereD(I) is the set of all infinitely differentiable functions with∈ compact support inI, see Appendix B) and integrating by parts, we find that, in the sense of distributions,

α,β α,β ψ�ω � = (g ψ)ω . (3.276) − − Next, we show that ψ ωα,β �is continuous� on[ 1,1] with( ψ ωα,β )( 1)=0. In- � − � ± deed, integrating (3.276) over any interval(x 1,x 2) [ 1,1], we obtain from the Cauchy–Schwarz inequality and (3.275) that ⊆ −

x2 α,β α,β α,β (ψ�ω )(x1) ( ψ�ω )(x2) (g ψ)ω dx − ≤ x | − | � 1 � x2 1/2 � x2 1/2 � α,β � α,β ω (x)dx g ψ ωα,β � ω (x)dx g ωα,β . ≤ x1 � − � x1 � � �� � �� � α,β α,β Hence, ψ�ω C[ 1,1]and( ψ�ω )( 1) are well-defined. Multiplying (3.276) by any function∈v H− 1 (I) and integrating± the resulting equality by parts, we ωα,β derive from (3.273∈) that

1 α,β 1 [ψ�ω v] =a α,β (ψ,v) (g,v) α,β =0, v H α,β (I). 1 − ω ∀ ∈ ω �− α,β � Hence,( ψ�ω )( 1�)=0. We are now ready± to prove (3.274). A direct computation from (3.276) leads to

2 1 ψ�� = (α + β)x+( α β) (1 x )− ψ� + (g ψ). (3.277) − − − − − One verifies readily� that � 2 ψ�� D 1 +D 2, (3.278) � �ωα,β ≤ whereD 1 =D 1(I1) +D 1(I2) withI 1 = ( 1,0) andI 2 = (0,1), and − 2 2 2 α 2,β 2 D1(Ij) =8( α + β ) ψ� ω − − dx,j=1,2, I | | � j 1 2 α,β D2 =2 (g ψ) ω dx . 1 − ��− � � � � � By (3.275), � � 2 2 D2 � g ψ � g . � − �ωα,β � � ωα,β α,β Thus, it remains to estimateD 1. Due to( ψ�ω )(1)=0, integrating ( 3.276) over (x,1) yields 1 α β α,β ψ� = (1 x) − (1+x) − (g ψ)ω dy. − x − � 3.5 Error Estimates for Polynomial Approximations 123

Plugging it intoD 1(I2) gives

1 1 2 α 2 β 2 α,β D1(I2)� (1 x) − − (1+x) − − (g ψ)ω dy dx 0 − x − � � 1 1 � 2 � α 2 α,β � (1 x) − − (g ψ)ω dy dx 0 − x − � � 1 � 1 � 2 α 1 α,β � (1 x) − (g ψ)ω dy dx. 0 − 1 x x − � � − � � Since α <1, using the Hardy inequality (B.39) leads to − 1 1 2 α,2β 2 α,β D1(I2)� (g ψ) ω dx� (g ψ) ω dx. 0 − 0 − � �

A similar inequality holds forD 1(I1). Therefore, a combination of the above esti- mates leads to

ψ�� α,β � g α,β , � �ω � � ω which, together with (3.275), implies (3.274). �� We are now in a position to derive the optimal estimate inL 2 -norm via the ωα,β duality argument. m 1 Theorem 3.37. Let α,β > 1.If ∂xu B − (I), then for1 m N+1, − ∈ α,β ≤ ≤ 1 π u u α,β � N,α,β − � ω (3.279) (N m+1)! (m+1)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 , ≤ � N! � � − − where c is a positive constant independent of m,N andu.

Proof. We have

1 (π u u,g) α,β 1 | N,α,β − ω | πN,α,β u u ωα,β = sup . (3.280) � − � 0=g L2 (I) g ωα,β � ∈ ωα,β � �

Let ψ be the solution to the auxiliary problem (3.273) for giveng L 2 (I). Taking ∈ ωα,β v= π1 u u in ( 3.273), we obtain from (3.269) that N,α,β − 1 1 (π u u,g) α,β =a (π u u, ψ) N,α,β − ω α,β N,α,β − =a (π1 u u, ψ π1 ψ). α,β N,α,β − − N,α,β 124 3 Orthogonal Polynomials and Related Approximation Results

Hence, by the Cauchy–Schwarz inequality, Theorem 3.36 and the regularity estimate (3.274), we have

1 1 1 (π u u,g) α,β π u u α,β π ψ ψ α,β | N,α,β − ω | ≤� N,α,β − � 1,ω � N,α,β − �1,ω 1 1 cN − π u u α,β ψ�� α+1,β+1 ≤ � N,α,β − � 1,ω � �ω 1 1 cN − π u u α,β g α,β . ≤ � N,α,β − � 1,ω � �ω Consequently, by (3.280),

1 1 1 π u u α,β cN − π u u α,β . � N,α,β − � ω ≤ � N,α,β − � 1,ω Finally, the desired result follows from Theorem 3.36. �� The approximation results in the Sobolev norms are of great importance for spec- tral approximation of boundary value problems. Oftentimes, it is necessary to take the boundary conditions into account and consider the projection operators onto the space of polynomials built in homogeneous boundary data. To this end, we assume that 1< α,β <1, and denote − 1 1 0 H (I)= u H (I):u( 1)=0 ,P = u P N :u( 1)=0 . 0,ωα,β ∈ ωα,β ± N ∈ ± � � � � If 1< α,β <1, then any function inH 1 (I) is continuous on[ 1,1], and there ωα,β holds− − 1 max u(x) � u 1,ωα,β , u H α,β (I). (3.281) x 1 | | � � ∀ ∈ ω | |≤ We leave the proof of this statement as an exercise (see Problem 3.22). Define

1 α,β ˆaα,β (u,v)= u�(x)v�(x)ω (x)dx, 1 �− which is the inner product ofH 1 (I), and induces the semi-norm, equivalent to 0,ωα,β the norm ofH 1 (I) (see Lemma B.7). 0,ωα,β Consider the orthogonal projection πˆ 1,0 :H 1 (I) P 0, defined by N,α,β 0,ωα,β → N ˆa (πˆ 1,0 u u,v)=0, v P 0. (3.282) α,β N,α,β − ∀ ∈ N The basic approximation result is stated as follows. 1 m 1 Theorem 3.38. Let 1< α,β <1.Ifu H (I) and ∂xu B − (I), then for − ∈ 0,ωα,β ∈ α,β 1 m N+1, ≤ ≤ πˆ 1,0 u u N,α,β − 1,ωα,β (3.283) � � (N m+1)! � � (1 m)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 , ≤ � N! � � − − where c is a positive constant independent of m,N andu. 3.5 Error Estimates for Polynomial Approximations 125

Proof. Let πα,β u betheL 2 - orthogonal projection as defined in (3.249). Setting N 1 ωα,β − x 1 α,β 1 α,β φ(x)= πN 1u� πN 1u�dη dξ , (3.284) 1 − − 2 1 − �− � �− � we have φ P 0, and ∈ N 1 α,β 1 α,β φ � = πN 1u� πN 1u�dη. − − 2 1 − �− Hence, by the triangle inequality,

1 α,β 1 α,β u� φ � ωα,β u � πN 1u� ωα,β + πN 1u�dη � − � ≤� − − � 2 1 − α,β ��− �ω (3.285) � α,β � � 1 � α,β �γ0 α,β � u � πN 1u� ωα,β + πN 1u�dη , ≤� − − � � 2 1 − ��− � � � α,β � � where γ0 is given in (3.109). Due tou( 1)= 0,� we derive from� the Cauchy– Schwarz inequality that for 1< α,β <1,± − 1 1 α,β α,β α, β α,β πN 1u�dx = (πN 1u� u �)dx γ0− − πN 1u� u � ωα,β . (3.286) 1 − 1 − − ≤ � − − � ��− � ��− � � � � � � � � � � Hence,� by definition� � and Theorem 3.35, �

1,0 α,β (πˆ u u) � φ � u � α,β c π u� u � α,β N,α,β − ωα,β ≤� − �ω ≤ � N 1 − �ω − (3.287) � (N �m+1)! � � (1 m)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 . ≤ � N! � � − − Finally, using the Poincare´ inequality (B.41) and (3.287) leads to

1,0 1,0 πˆ u u α,β c ( πˆ u u) � α,β � N,α,β − � ω ≤ � N,α,β − �ω (3.288) (N m+1)! (1 m)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 . ≤ � N! � � − − This completes the proof. �� As in the proofof Theorem 3.37,we can derivean optimal estimate for πˆ 1,0 u N,α,β − u in theL 2 -norm by using a duality argument. One may refer to Canuto et al. ωα,β (2006) for the Legendre and Chebyshev cases, and to Guo and Wang (2004) for the general cases. Moreover, we shall introduce in Chap.5 a family of generalized Jacobi polynomials, and a concise analysis based on this notion will automatically lead to the desired results. When we apply the Jacobi approximation (e.g., the Chebyshev approximation) to boundary-value problems, it is often required to use the projection operator associated with the bilinear form 126 3 Orthogonal Polynomials and Related Approximation Results

1 α,β aα,β (u,v)= ∂xu(x)∂x v(x)ω (x) dx, (3.289) 1 �− � � which is closely related to the weighted Galerkin formulation for the model equation

u��(x)+ μu(x)=f(x), μ 0;u( 1)=0. − ≥ ± Incontrast with (3.282), we define the orthogonalprojection π1,0 :H 1 (I) N,α,β 0,ωα,β → 0 PN, such that a (u π1,0 u,v)=0, v P 0. (3.290) α,β − N,α,β ∀ ∈ N The bilinear form is continuous and coercive as stated in the following lemma. Lemma 3.5.If 1< α,β <1, then for any u,v H 1 (I), − ∈ 0,ωα,β

a (u,v) C 1 u α,β v α,β , (3.291) | α,β | ≤ | |1,ω | |1,ω and 2 a (v,v) C 2 v , (3.292) α,β ≥ | |1,ωα,β where C1 and C2 are two positive constants independent of u and v.

Proof. Since 1< α,β <1, we have from (B.40) that − α,β a (u,v) (u �,v �) α,β + (u�,v(ω )�) | α,β | ≤ | ω | u α,β v α,β +2 u α,β v α 2,β 2 ≤ | |1,ω | |1,ω | | 1,ω � �ω − − C 1 u α,β v α,β . ≤ | |1,ω | |1,ω We now prove the coercivity. A direct calculation gives

2 1 2 aα,β (v,v)= v α,β + v ,Wα,β α 2,β 2 , | | 1,ω 2 ω − − � � where

W (x)=(α + β)(1 α β)x2 α,β − − +2( α β)(1 α β)x+ α + β ( α β)2. − − − − −

By the property of quadratic polynomials, one verifies readily thatW α,β (x) 0, provided that ≥ (α + β)(α + β 1) 0, − ≥ W ( 1)= 4 β(β 1) 0, ⎧ α,β − − − ≥ ⎨⎪W (1)= 4 α(α 1) 0, α,β − − ≥ ⎩⎪ 3.5 Error Estimates for Polynomial Approximations 127

or

(α + β)(α + β 1) 0, − ≤ 2 2 2 �4(α β) (α + β 1) +4( α + β)(α + β 1)( α + β ( α β) ) 0. − − − − − ≤ If 0 α,β 1, then both of them are valid, which implies ( 3.292) with 0 α, β ≤1. ≤ ≤ ≤Next, let 1< α,β <0andu(x)= ωα,β (x)v(x).As0< α, β <1, it follows from the above− shown case that − −

2 aα,β (v,v)=a α, β (u,u) u α, β . (3.293) − − ≥| | 1,ω− − On the other hand, by (B.40),

2 2 2 2 2 2 v α,β 2 u α, β +8( α + β ) u α 2, β 2 c u α, β . (3.294) | |1,ω ≤ | | 1,ω− − � �ω− − − − ≤ | | 1,ω− − A combination of (3.293) and (3.294) leads to (3.292) with 1< α,β <0. Now, let 1< α 0 β <1andu(x)=(1 x) α v(x). −We deduce from Corol- − 1 ≤ ≤ − lary B.1 thatu H α,0 (I), so by (B.40), ∈ 0,ω− 2 α 2 2 2 2 v α,β = (1 x) − u α,β 2 u α,β +2 α u α 2,β | |1,ω | − |1,ω ≤ | | 1,ω− || ||ω− − 2 2 2 2 2 u α,β +8 α u α 2,β 2 c u α,β . ≤ | | 1,ω− � �ω− − − ≤ | | 1,ω− In view of 1< α 0 β <1, we have − ≤ ≤ 2 2 2 2 u α,β u α,β 2 α(α +1) u α 2,β +2 β(1 β) u α,β 2 | |1,ω− ≤ | |1,ω− − � � ω− − − � �ω− − α β = ∂x((1 x) − u),∂x((1+x) u) =a (v,v). − α,β This leads to (3.292�) with 1< α 0 β <1. � We can treat the remaining− case≤ 1≤< β 0 α < 1 in the same fashion as above. − ≤ ≤ �� 1 m 1 Theorem 3.39. Let 1< α,β <1.Ifu H (I) and ∂xu B − (I), then for − ∈ 0,ωα,β ∈ α,β 1 m N+1 and μ =0,1, ≤ ≤ u π1,0 u − N,α,β μ,ωα,β (3.295) � N� m 1 ! � (� + ) μ (m+1)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 , ≤ � N! � � − − where c is a positive constant independent of m,N andu.

ˆ 1,0 Proof. We first prove the case μ =1. Let πN,α,β be the projection operator defined in (3.282). By the definition (3.290),

a (π1,0 u u, π1,0 u u)=a (π1,0 u u, πˆ 1,0 u u), α,β N,α,β − N,α,β − α,β N,α,β − N,α,β − 128 3 Orthogonal Polynomials and Related Approximation Results

which, together with Lemma 3.5, gives

π1,0 u u 2 c a (π1,0 u u, πˆ 1,0 u u) | N,α,β − | 1,ωα,β ≤ | α,β N,α,β − N,α,β − | 1,0 1,0 c π u u α,β πˆ u u α,β . ≤ | N,α,β − | 1,ω | N,α,β − | 1,ω Hence, the estimate (3.295) with μ = 1 follows from Theorem 3.38 and the inequal- ity (B.41). To prove the case μ =0, we resort to the duality argument. Giveng 2 1 ∈ L α 1,β 1 (I), we consider a auxiliary problem. It is to findv H α,β (I) such ω − − ∈ 0,ω that 1 aα,β (v,z) = (g,z) α 1,β 1 , z H α,β (I). (3.296) ω − − ∀ ∈ 0,ω Since by (B.40),

(g,z) α 1,β 1 c g α 1,β 1 z α 2,β 2 | ω − − | ≤ � � ω − − � �ω − − c g α 1,β 1 z α,β , ≤ � � ω − − | |1,ω we deduce from Lemma 3.5 and the Lax-Milgram lemma (see Chap.1 or AppendixB) that the problem(3.296)has a uniquesolution inH 1 (I). Moreover, 0,ωα,β in the sense of distributions, we havev (x)= (1 x 2) 1g(x). Therefore, �� − − −

v α+1,β+1 = g α 1,,β 1 . | |2,ω � � ω − − Takingz= π1,0 u u in ( 3.296), we obtain from Lemma 3.5 and Theorem 3.39 N,α,β − that

1,0 1,0 (g,π u u) α 1,β 1 = a α,β (v,π u u) | N,α,β − ω − − | | N,ω − | = a (π1,0 v v, π1,0 u u) | α,β N,α,β − N,α,β − | 1,0 1,0 c π v v α,β π u u α,β ≤ | N,α,β − | 1,ω | N,α,β − | 1,ω 1 1,0 cN − v α+1,β+1 π u u α,β ≤ | |2,ω | N,α,β − | 1,ω (N m+1)! (m+1)/2 m c − (N+m) − g ωα 1,,β 1 ∂x u ωα+m 1,β+m 1 . ≤ � N! � � − − � � − − Consequently,

1,0 (π u u,g) α 1,β 1 1,0 N,α,β ω − − π u u α 1,β 1 = sup | − | N,α,β ω − − � − � 0 g L2 I g ωα 1,,β 1 = α 1,β 1 ( ) − − � ∈ ω − − � �

(N m+1)! (m+1)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 . ≤ � N! � � − − It is clear that 1,0 1,0 π u u α,β c π u u α 1,β 1. � N,α,β − � ω ≤ � N,α,β − � ω − − Thus, the desired result follows. �� 3.5 Error Estimates for Polynomial Approximations 129 3.5.3 Interpolations

This section is devoted to the analysis of polynomial interpolation on Jacobi-Gauss- type points. The analysis essentially relies on the polynomial approximation re- sults derived in the previous section, and the asymptotic properties of the nodes and weights of the associated quadrature formulas. For clarity of presentation, we start with the Chebyshev-Gauss interpolation. Re- call the Chebyshev-Gauss nodes and weights (see Theorem 3.30):

2j+1 π x j =cos π, ω j = ,0 j N. 2(N+1) N+1 ≤ ≤

2 1/2 To this end, we denote the Chebyshev weight function by ω = (1 x )− . − c An essential step is to show the stability of the interpolation operatorI N . 1 Lemma 3.6. For anyu B 1/2, 1/2(I), we have ∈ − −

c π 2 1/2 I u ω u ω + (1 x ) u� ω . (3.297) � N � ≤� � N+1 � − � Proof. Letx= cos θ and ˆuθ( ) =u(cos θ). Thanks to the exactness of the Chebyshev-Gauss quadrature (cf. (3.217)), we have

N N c 2 c 2 π 2 π 2 INu ω = I Nu N,ω = ∑ u (x j) = ∑ ˆu(θ j), � � � � N+1 j=0 N+1 j=0 where 2j+1 θ j = arccos(xj) = π,0 j N. 2(N+1) ≤ ≤

Denote jπ a j = ,0 j N+1. N+1 ≤ ≤ It is clear that

θ j K j := [a j,a j 1],0 j N, ∈ + ≤ ≤ and the length of the subinterval is K j = π/(N+1). Applying the embedding | | inequality (B.34) onK j yields

N+1 π ˆu max ˆu ˆu 2 ˆu 2 (θ j) (θ) L (Kj) + ∂θ L (Kj). | | ≤ θ Kj | | ≤ π � � N+1 � � ∈ � � 130 3 Orthogonal Polynomials and Related Approximation Results

Hence,

N c π I u ω ˆu(θ j) � N � ≤ N+1 ∑ | | � j=0 N π ˆu 2 + ∂ ˆu 2 ∑ L (Kj) θ L (Kj) ≤ j=0 � � N+1 � � � � π ˆu 2 + ∂ ˆu 2 . ≤� �L (0,π) N+1 � θ �L (0,π) Finally, the inverse change of variable θ x leadsto ( 3.297). → �� Now, we are in a position to present the main result on the Chebyshev-Gauss interpolation error estimates. Theorem 3.40. Foranyu B m (I) with m 1, we have that for any0 l ∈ 1/2, 1/2 ≥ ≤ ≤ m N+1, − − ≤ l c ∂x(IN u u) l 1/2,l 1/2 � − � ω − − (3.298) (N m+1)! l (m+1)/2 m c − (N+m) − ∂x u ωm 1/2,m 1/2 , ≤ � N! � � − − where c is a positive constant independent m,N andu.

c 1/2, 1/2 Proof. Let πN := πN− − be the Chebyshev orthogonal projection operator de- c c c c fined in (3.249). Since πNu P N, we haveI N (πNu)= πN u. Using Lemma 3.6 and Theorem 3.35 with α = β =∈ 1/2 leads to − c c c c I u π u ω = I (u π u) ω � N − N � � N − N � c 1 c c u π u ω +N − ∂x(u π u) 1 ≤ � − N � � − N �ω− � (N m+1)! (m+1)/2 m � c − (N+m) − ∂x u ωm 1/2,m 1/2 , ≤ � N! � � − − which, together with the inverse inequality (3.236), leads to

l c c l c c ∂x(IN u πNu) l 1/2,l 1/2 cN INu πNu ω � − �ω − − ≤ � − � (N m+1)! l (m+1)/2 m c − (N+m) − ∂x u ωm 1/2,m 1/2 . ≤ � N! � � − − Finally, it follows from the triangle inequality and Theorem 3.35 that

l c l c c ∂x(IN u u) l 1/2,l 1/2 ∂x(INu πNu) l 1/2,l 1/2 � − � ω − − ≤� − �ω − − l c + ∂x(πN u u) l 1/2,l 1/2 � − � ω − − (N m+1)! l (m+1)/2 m c − (N+m) − ∂x u ωm 1/2,m 1/2 . ≤ � N! � � − − This ends the proof. �� 3.5 Error Estimates for Polynomial Approximations 131

We observe that the Chebyshev-Gauss interpolation shares the same optimal 1/2, 1/2 order of convergence with the orthogonal projection πN− − (cf. Theorem 3.35). Next, we extend the above argument to the general Jacobi-Gauss-type interpola- tions. An essential difference is that unlike the Chebyshev case, the explicit expres- sions of the nodes and weights are not available. Hence, we have to resort to their asymptotic expressions. N Let x j,ω j be the set of Jacobi-Gauss, Jacobi-Gauss-Radau, or Jacobi- { } j=0 Gauss-Lobatto nodes and weights relative to the Jacobi weight function ωα,β (cf. N Sect. 3.2). Assume that x j j=0 are arranged in descending order, and set θ j = N { } { arccos(xj) j=0. For the variable transformationx= cos θ,θ [0, π],x [ 1,1], it is clear that} ∈ ∈ − dθ 1 θ 2 θ 2 = ,1 x=2 sin ,1+x=2 cos . (3.299) dx − √1 x 2 − 2 2 − � � � �

3.5.3.1 Jacobi-Gauss Interpolation

Recall the asymptotic formulas of the Jacobi-Gauss nodes and weights given by Theorem 8.9.1 and Formula (15.3.10) of Szego¨ (1975). Lemma 3.7. For α,β > 1, we have − 1 1 θ j = cos− x j = (j+1) π +O(1) , (3.300) N+1 � � with O(1) being uniformly bounded for all values j=0,1,...,N, and

α+β +1 2α+1 2β +1 2 π θ j θ j ω j = sin cos ,0 j N. (3.301) ∼ N+1 2 2 ≤ ≤ � � � � As with Lemma 3.6, we first show the stability of the Jacobi-Gauss interpolation α,β operatorI N . Lemma 3.8. For any α,β > 1,andanyu B 1 (I), − ∈ α,β α,β 1 I u α,β � u α,β +N − u� α+1,β+1 . (3.302) � N �ω � � ω � �ω Proof. Letx= cos θ and ˆuθ( ) =u(x) with θ (0, π). By the exactness of the Jacobi-Gauss quadrature (cf. Theorem 3.25) and∈ Lemma 3.7,

N Iα,β u 2 I α,β u 2 u2 x N ωα,β = N N,ωα,β = ∑ ( j)ω j � � � � j=0 N 2α+1 2β +1 θ j θ j �N 1 ˆu2(θ ) sin cos . − ∑ j 2 2 j=0 � � � � 132 3 Orthogonal Polynomials and Related Approximation Results

The asymptotic formula (3.300) implies that θ j K j [a 0,a 1] (0, π), wherea 0 = O(1) Nπ+O(1) ∈ ⊂ ⊂ c N+1 ,a 1 = N+1 and the length of each closed subintervalK j is N+1 . Hence,

N 1 1 α,β 1 θ α+ 2 θ β + 2 2 IN u ωα,β �N − ∑ max ˆuθ( ) sin cos . � � θ Kj 2 2 j=0 ∈ � � � � � � � � � � For notational simplicity, we denote � �

1 1 θ α+ 2 θ β + 2 χα,β (θ) = sin cos . 2 2 � � � � Applying the embedding inequality (B.34) onK j yields

N α,β α,β 1 α,β I u α,β � ˆuχ 2 +N − ∂ ˆuχ 2 N ω ∑ L (Kj) θ L (Kj) � � j=0 � � α�,β � 1 � α�,β �� � ˆuχ � 2 � +N − ∂θ �ˆuχ 2 � L (0,π) L (a0,a1) � � ˆuχα,β � +N 1�χα�,β ∂ ˆu�� � �L2(0,π) − � θ L�2(0,π) 1 α 1,β 1 +� N − �ˆuχ − − 2� . � � � L �(a0,a1) � � � In view of (3.299), an inverse change� of variable� leads to

π 2 θ 2α+1 θ 2β +1 ˆuχα,β = ˆu2(θ) sin cos dθ L2(0,π) 2 2 �0 � � 1 � � � � 1 � � � u2(x)(1 x) α+1/2(1+x) β +1/2 dx 1 − √1 x 2 �− − � u 2 , � � ωα,β and similarly,

α,β χ ∂ ˆu � ∂xu α+1,β+1 . θ L2(0,π) � �ω � � We treat the last term as� �

1 α 1,β 1 1 α,β N− ˆuχ − − L2(a ,a ) � sup ˆuχ L2(a ,a ) 0 1 a θ a N sinθ 0 1 � 0≤ ≤ 1 � � � α,β � � � � � ˆuχ � �u α,β�, L2(0,π) � � ω 1 � � 1 where due to the facta 0 =O(N ) and�a 1 = π� O(N ), we have − − − 1 sup c. a θ a N sinθ ≤ 0≤ ≤ 1 A combination of the above estimates leads to the desired result. �� 3.5 Error Estimates for Polynomial Approximations 133

As a consequence of Lemma 3.8, we have the following inequality in the polynomial space.

Corollary 3.7. For any φ PM and ψ P L, ∈ ∈ α,β M I φ α,β � 1+ φ α,β , (3.303a) � N �ω N � �ω M � L � φ,ψ α,β � 1+ 1+ φ α,β ψ α,β . (3.303b) |� �N,ω | N N � �ω � �ω � �� � Proof. Using the inverse inequality (3.236) and (3.302) gives

α,β 1 M I φ α,β � φ α,β +N − ∂xφ α+1,β+1 � 1+ φ α,β . � N �ω � �ω � �ω N � �ω � � Therefore,

α,β α,β (3.150) α,β α,β φ,ψ α,β = I φ,I ψ α,β = (I φ,I ψ) α,β |� �N,ω | |� N N �N,ω | | N N ω | M L � 1+ 1+ φ α,β ψ α,β . N N � �ω � �ω � �� � This ends the proof. �� With the aid of the stability result (3.302), we can estimate the Jacobi-Gauss interpolation errors by using an argument similar to that for Theorem 3.40. Theorem 3.41. Let α,β > 1. For any u B m (I) with m 1, we have that for − ∈ α,β ≥ 0 l m N+1, ≤ ≤ ≤ ∂ l(Iα,β u u) x N − ωα+l,β+l (3.304) � (�N m+1)! l (m+1)/2 m � c � − (N+m) − ∂x u ωα+m,β+m , ≤ � N! � � where c is a positive constant independent of m,N andu. Similar to (3.267), the Jacobi-Gauss interpolation errors measured in the norms of the usual Sobolev spacesH l (I)(l 1) are not optimal. For instance, a standard ωα,β argument using (3.302), Theorem 3.34≥, and Theorem 3.36 leads to that for anyu Bm (I) with1 m N+1, ∈ α,β ≤ ≤ α,β I u u α,β � N − � 1,ω (3.305) (N m+1)! (3 m)/2 m c − (N+m) − ∂x u ωα+m 1,β+m 1 . ≤ � N! � � − − Now, we consider the Jacobi-Gauss-Radau and Jacobi-Gauss-Lobatto interpolations. 134 3 Orthogonal Polynomials and Related Approximation Results

3.5.3.2 Jacobi-Gauss-Radau Interpolation

N In view of (3.140), theN interior Jacobi-Gauss-Radau nodes x j turn out to { } j=1 be the Jacobi-Gauss nodes with the parameter( α,β +1). Hence, by ( 3.300) and (3.301), 1 θ j = arccos(xj) = jπ +O(1) ,1 j N, (3.306) N ≤ ≤ and � �

α+β +2 2α+1 2β +3 2 π 1 θ j θ j ω j = sin cos ∼ N 1+x j 2 2 � � � � (3.307) α+β +1 2α+1 2β +1 2 π θ j θ j = sin cos ,1 j N. ∼ N 2 2 ≤ ≤ � � � � Moreover, applying the Stirling’s formula (A.7) to (3.134a) yields

2β 2 ω0 =O N− − . (3.308)

Similar to Lemma 3.8, we have the following� stability� of the Jacobi-Gauss-Radau interpolation operator. Lemma 3.9. For anyu B 1 (I), ∈ α,β

α,β β 1 1 I u α,β �N − − u( 1) + u α,β +N − u α+1,β+1 . (3.309) � N �ω | − | � � ω | |1,ω Proof. By the exactness of the Jacobi-Gauss-Radau quadrature (cf. Theorem 3.26),

N Iα,β u 2 I α,β u 2 u 2 1 u2 x N ωα,β = N N,ωα,β = ( )ω0 + ∑ ( j)ω j. � � � � − j=1

Thanks to (3.306) and (3.307), using the same argument as for Lemma 3.8 leads to

N u2 x u 2 N 2 u 2 ∑ ( j)ω j � ωα,β + − 1,ωα+1,β+1 . j=1 � � | |

Hence, a combination of the above two results and (3.308) yields (3.309). �� As a direct consequence of Lemma 3.9, we have the following results.

Corollary 3.8. For any φ PM and ψ P L with φ( 1)= ψ( 1)=0, ∈ ∈ − − α,β M I φ α,β � 1+ φ α,β , (3.310a) � N �ω N � �ω M � L � φ,ψ α,β � 1+ 1+ φ α,β ψ α,β . (3.310b) |� �N,ω | N N � �ω � �ω � �� � In order to deal with the boundary term in Lemma 3.9, we need to estimate the projection errors at the endpoints. 3.5 Error Estimates for Polynomial Approximations 135

Lemma 3.10. Let α,β > 1.For u B m (I), − ∈ α,β if α +1

∞ α,β α,β α,β (πN u u)(1) ∑ ˆun Jn (1) − ≤ n=N+1 | || | � � 1/2 1/2 � ∞� ∞ α,β 2 α,β 1 α,β 2 α,β ∑ Jn (1) (hn,˜m)− ∑ ˆun hn,˜m ≤ �n=N+1 | | � �n=N+1 | | � 1/2 ∞ α,β 2 α,β 1 ˜m ∑ Jn (1) (hn,˜m)− ∂x u ωα+˜mβ, +˜m. ≤ �n=N+1 | | � � �

By (3.94), (3.254) and the Stirling’s formula (A.7), we find

α,β 2 1+2α Jn (1) ˜m(n ˜m)! n | | ce − − , n N+1 1. α,β ≤ n! n ˜m ∀ ≥ � hn,˜m

Moreover, by (A.8) and the inequality: 1 x e x forx [0,1], − ≤ − ∈ ˜m n ˜m+1/2 ˜m(n ˜m)! e− ˜m − ˜m e− − c 1 cn − . n! ≤ n ˜m − n ≤ � � Hence, for ˜m>α +1,

∞ α,β 2 ∞ ∞ Jn (1) 2α+1 2˜m 2α+1 2˜m c 2(1+α ˜m) ∑ | | c ∑ n − c x − dx N − . α,β ≤ ≤ N ≤ ˜m n=N+1 hn,˜m n=N+1 � A combination of the above estimates leads to

α,β c 1+α ˜m ˜m (π u u)(1) N − ∂ u α+˜mβ, +˜m,˜m> α +1. (3.313) | N − | ≤ √ ˜m � x �ω This gives (3.311). Thanks to (3.105), we derive (3.312) easily. �� 136 3 Orthogonal Polynomials and Related Approximation Results

Thanks to Lemma 3.9, Lemma 3.10 and Theorem 3.35, we can derive the following result by an argument analogous to that for Theorem 3.40. Theorem 3.42. For α,β > 1andanyu B m (I), we have that for0 l mand − ∈ α,β ≤ ≤ β +1

l α,β (N m+1)! l (m+1)/2 m (3.314) ∂x(IN u u) ωα+l,β+l c − N − ∂x u ωα+m,β+m , − ≤ � N! � � � � where� c is a positive� constant independent m,N andu.

3.5.3.3 Jacobi-Gauss-Lobatto Interpolation

N 1 The relation (3.141) indicates that theN 1 interior JGL nodes x j − are the JG − { } j=1 nodes with the parameter( α +1, β +1). Hence, by ( 3.300), 1 θ j = arccos(xj) = jπ +O(1) ,1 j N 1. (3.315) N 1 ≤ ≤ − − � � Moreover, we find from (3.141) and (3.301) that the associated weights have the asymptotic property:

α+β +1 2α+1 2β +1 2 π θ j θ j ω j = sin cos ,1 j N 1. (3.316) ∼ N 1 2 2 ≤ ≤ − − � � � � Furthermore, applying the Stirling’s formula (A.7) to the boundary weights in (3.139a) and (3.139b) yields

2β 2 2α 2 ω0 =O N− − , ωN =O N− − .

Hence, similar to Lemmas� 3.8 and� 3.9, we can� derive� the following stability result. Lemma 3.11. Forany u B 1 (I), ∈ α,β

α,β α 1 β 1 I u α,β �N − − u(1) +N − − u( 1) � N �ω | | | − | (3.317) 1 + u α,β +N − u α+1,β+1 . � � ω | |1,ω As with Corollaries 3.7 and 3.8, the following bounds can be obtained directly from Lemma 3.11.

Corollary 3.9. For any φ PM and ψ P L with φ( 1)= ψ( 1)=0, ∈ ∈ ± ± α,β M I φ α,β � 1+ φ α,β , (3.318a) � N �ω N � �ω M � L � φ,ψ α,β � 1+ 1+ φ α,β ψ α,β . (3.318b) |� �N,ω | N N � �ω � �ω � �� � Problems 137

Similar to the Jacobi-Gauss-Radau case, we can derive the following estimates by using Lemmas 3.11 and 3.10, and Theorem 3.35. Theorem 3.43. For α,β > 1,andanyu B m (I), we have − ∈ α,β

l α,β (N m+1)! l (m+1)/2 m ∂x(IN u u) ωα+l,β+l c − N − ∂x u ωα+m,β+m , (3.319) − ≤ � N! � � � � for�0 l m and max� α +1, β +1 α +1 and/orm> β +1 for the Radau and Lobatto interpolations. As a result, for the Legendre-Gauss-Lobatto interpola- tion, the estimate stated in Theorem 3.43 does not hold form=1. In Chap.5 (see Sect.6.5), we shall take a different approach to derive the follow- ing estimate for the Legendre-Gauss-Lobatto interpolation.

m Theorem 3.44. Forany u B 1, 1(I), we have that for1 m N+1, ∈ − − ≤ ≤

∂x(IN u u) +N I Nu u 1, 1 � − � � − � ω− − (3.320) (N m+1)! (1 m)/2 m c − (N+m) − ∂ u m 1,m 1 , x ω − − ≤ � N! � � where c is a positive constant independent of m,N andu.

Problems

3.1. Derive the properties stated in Corollary 3.2.

3.2. Let p n be a sequence of orthogonal polynomials defined on a finite interval { } (n) (n) (a,b), and letx n be the largest zero ofp n. Show that limn ∞ xn exists. → N 3.3. Regardless of the choice of x j,ω j j=0, the quadrature formula (3.33) cannot have degree of precision greater{ than 2N}+1.

3.4. Let

T t p x S s 1 p x = n j := n( j) 0 n,j N , = jn := γn− n( j)ω j 0 n,j N ≤ ≤ ≤ ≤ � � � � 1 be the transform matrices associated with (3.62) and (3.64). Show thatT=S − . 138 3 Orthogonal Polynomials and Related Approximation Results

3.5. For α > ρ > 1 and β > 1, show that − − 1 β +ρ+1 α,β ρ,β 2 Γ (ρ +1)Γ (n+ β +1) Jn (x)ω (x)dx= . 1 n!Γ (α ρ)Γ (ρ + β +n+2) �− − 3.6. Prove the following Rodrigues-like formula:

( 1)m(n m)! (1 x) α(1+x) β Jα,β (x)= − − − n 2mn! × m α+m β +m α+m,β +m (3.321) ∂x (1 x) (1+x) Jn m (x) , − − α�,β > 1,n m 0. � − ≥ ≥ 3.7. Derive the formulas in Theorem 3.20. 3.8. Derive the formulas in Theorem 3.27. 3.9. Prove that the following equation holds for integersn>m,

m m d 2 m d Ln m+1 (1 x ) + ( 1) λm,nLn =0, (3.322) dxm − dxm − � � where (n+m)! λm n = . (3.323) , (n m)! − 3.10. Prove the orthogonality

1 (m) (m) 2 m 2 Ln (x)Lk (x)(1 x ) dx= λm,n Ln δnk. 1 − � � �− 3.11. Show that the Legendre polynomials satisfy

m n m (n+m)! ∂ Ln( 1)=( 1) − . (3.324) x ± ± 2mm!(n m)! − 3.12. Let 2 A φ = ∂x((1 x )∂xφ) − − be the Sturm-Liouville operator. Verify that

∂ k(A φ) = (1 x 2)∂ k+2φ +2(k+1)x ∂ k+1φ +k(k+1) ∂ kφ. x − − x x x Hence, we have k ∂ (A φ) � φ k 2. � x � � � + 3.13. Givenu P N, we consider the expansions ∈ N k (k) ∂x u(x)= ∑ ˆun Ln(x),0 k

Prove the following relations:

N (1) (0) ˆun = (2n+1) ∑ ˆup ; p=n+1 n+podd N (1) (2n+1) (0) ˆun = ∑ p(p+1) n(n+1) ˆup ; 2 p=n+2 − n+p even� � 1 (k) 1 (k) (k 1) ˆu ˆu =ˆun − . 2n 1 n 1 − 2n+3 n+1 − − 3.14. Prove that 0, ifn odd, Ln(0)= 2 ⎧n!2 n (n/2)! − , ifn even. ⎨ − � � 3.15. According to the formula⎩ (4.8.11) of Szego¨ (1975),we have that for anyn N andx [ 1,1], ∈ ∈ − π 2 n Ln(x)= x+i 1 x cosθ dθ, 0 − � � where i= √ 1. Prove that the Legendre� polynomials� are uniformly bounded be- tween the parabolas−

1+x 2 1+x 2 L n(x) , x [ 1,1]. − 2 ≤ ≤ 2 ∀ ∈ −

3.16. Givenu P N, we consider the expansions ∈ N k (k) ∂x u(x)= ∑ ˆun Tn(x),0 k

N (1) 2 (0) ˆun = ∑ pˆup ; cn p=n+1 n+podd N (2) 1 2 2 (0) ˆun = ∑ p p n ˆup ; cn p=n+2 − n+p even � � N (3) 1 2 2 2 2 2 2 (0) ˆun = ∑ p p (p 2) 2p n + (n 1) ˆup ; 4cn p=n+3 − − − n+p odd � � N (4) 1 2 2 2 4 2 2 4 2 2 2 (0) ˆun = ∑ p p (p 4) 3p n +3p n n (n 4) ˆup , 24cn p=n+4 − − − − n+p even � � 140 3 Orthogonal Polynomials and Related Approximation Results

and the recurrence formula

(k) (k) (k 1) cn 1 ˆun 1 ˆun+1 =2nˆun − . − − − 3.17. Show that m 2 2 m n+m n k ∂x Tn( 1)=( 1) ∏ − . ± ± k=0 2k+1 3.18. Prove that

1 2 2 1 Tn(x) dx=1 (4n 1) − ,n 0. 1 − − ≥ �− � � 3.19. Show that:

(a) The constants α j and β j in (3.25) are the same as the coefficients in (3.7). (b) The characteristic polynomial of the matrixA n+1 is the monic polynomial ¯pn+1(x), namely,

¯pn 1(x)= det(xIn 1 A n 1),n 1, (3.325) + + − + ≥− 3.20. Prove the inverse inequalities

α φ �N φ α,α , φ P N, α 0, � � � �ω ∀ ∈ ≥ and

φ 1, 1 �N φ , φ PN, φ( 1)=0. � �ω− − � � ∀ ∈ ± 3.21. Prove Theorem 3.34 for the Chebyshev case, that is, for any φ P N, ∈ 2 1 ∂xφ ω �N φ ω , ω(x)= . � � � � √1 x 2 − 3.22. Show that for 1< α,β <1, we haveH 1 (I) C( I¯)and( 3.281) holds. − ωα,β ⊆ 3.23. LetI N be the Legendre-Gauss interpolation operatorN+1 Legendre-Gauss- Lobatto points. Verify that foru=L N+1 L N 1, − − 1/2 INu u 1 cN u� . � − � H ≥ � �

3.24. LetI N be the interpolation operator onN+1 Legendre-Gauss-Lobatto points. Show that for anyu H 1(I), ∈ 0 1 INu 1, 1 c u 1, 1 +N − ∂xu . (3.326) � �ω− − ≤ � �ω− − � � � �