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1
(q) (q’1) (q)
k = N ’ 1, N ’ 2, N ’ 3, . . . , 1
ak’1 = 2 k ak + ak+1 ,
ck’1

where ck = 2 if k = 0 and ck = 1 for k > 0.

Notes:

d
(i) (A.16)
Tn (x) = n Un’1 (x)
dx
Higher order differentiation is mildly ill-conditioned as discussed by Breuer and Ever-
son (1992), Huang and Sloan (1992), Merry¬eld and Shizgal (1993) and Bayliss, Class and
Matkowsky (1994). Heinrichs (1991) has suggested a partial remedy: Basis recombination.

Integration:

x
Tn+1 (x) Tn’1 (x)
n≥2 (A.17)
dy Tn (y) = + ,
2(n ’ 1)
2(n + 1)

Flyer (1998) gives asymptotic upper bounds for the coef¬cients in the Chebyshev series
for the iterated integral of arbitrary order as well as a discussion of integration identities in
general.
Karageorghis (1988a, b) has analyzed the Chebyshev coef¬cients of the general order
derivative and of its moments.


Collocation Points: either of the two choices

(2i ’ 1)π [“Roots” or
(A.18)
xi = cos i = 1, . . . , N
“Gauss-Chebyshev”]
2N
πi [“Extrema-plus-Endpoints”
i = 0, . . . , N ’ 1 (A.19)
xi = cos
N ’1 or “Gauss-Lobatto”]

Differential Equation (Sturm-Liouville Problem)


d2 Tn dTn
(1 ’ x ) ’x
2
+ n2 Tn = 0, (A.20)
n = 0, 1, 2, . . .
2
dx dx
A.3. CHEBYSHEV POLYNOMIALS OF THE SECOND KIND: UN (X) 499

A.3 Chebyshev Polynomials of the Second Kind: Un (x)


Applications: Proportional to ¬rst derivative of Chebyshev polynomials of the ¬rst
kind. Rarely used as a basis set.

x ∈ [’1, 1]
Interval:

Weight:


ρ(x) ≡ 1 ’ x2 (A.21)

Inner Product:

1
π
1 ’ x2 dx = (A.22)
Um (x) Un (x) δmn
2
’1

Standardization:

(A.23)
Un (1) = n + 1

Trigonometric Form:

sin[(n + 1)θ]
Un (cos θ) ≡ (A.24)
sin θ
Inequality:

|Un+1 (x)| ¤ n + 1 for all n, all x on [-1, 1] (A.25)

Three-Term Recurrence and Starting Values:

U0 (x) ≡ 1 U1 (x) ≡ 2 x (A.26)
;
Un+1 (x) = 2 x Un (x) ’ Un’1 (x) n≥1 (A.27)

Differentiation Rules:


(i) Use trigonometric form (A.24)
(ii) Use relationship between Chebyshev & Gegenbauer polynomials

dm (m+1)
Un (x) = 2m m! Cn’m (x) (A.28)
dxm
Collocation Points:


xi = ’ cos (A.29)
; i = 1, 2, . . . , N
N +1
APPENDIX A. A BESTIARY OF BASIS FUNCTIONS
500

A.4 Legendre Polynomials: Pn (x)


Applications: Spectral elements; alternative to Chebyshev polynomials for non-periodic
problems; also the axisymmetric spherical harmonics for problems in spherical geometry.
x ∈ [’1, 1] ρ(x) ≡ 1
Interval: Weight:
Inner Product:

1
2
(A.30)
Pm Pn dx = δmn
2n + 1
’1

Endpoint Values:
Pn (±1) = (±1)n , dPn /dx(±1) = (±1)n’1 n(n + 1)/2 (A.31)
Inequalities:

|Pn (x)| ¤ 1 |dPn /dx| ¤ n(n + 1)/2 for all n, all x ∈ [’1, 1] (A.32)
Nonuniformity:

1 · 3 · · · (n ’ 1)
∼ n even (A.33)
Pn (0) = 2/(π n),
2 · 4 · · · (n)
Asymptotic Approximation as n ’ ∞ for ¬xed t:

2 π
+ O(n’3/2 )
Pn (cos(t)) ∼ (A.34)
sin (n + 1/2)t +
nπ sin(t) 4



Three-Term Recurrence and Starting Values:

P0 ≡ 1, (A.35)
P1 (x) = x

(n + 1) Pn+1 (x) = (2 n + 1) x Pn (x) ’ n Pn’1 (x) (A.36)
Differentiation:

Use relationship between Legendre and Gegenbauer Polynomials
dm (m+1/2)
P (x) = 1 · 3 · 5 · · · (2m ’ 1) Cn’m (x) (A.37)
mn
dx
Phillips(1988) gives the Legendre coef¬cients of a general-order derivative of an in-
¬nitely differentiable function.
Collocation Points (Gaussian Quadrature Abscissas):

Not known in closed form for general N ; see pgs. 916“919 of Abramowitz & Ste-
gun(1965), NBS Handbook. Analytical expressions for up to 9-point grids are given in Ap-
pendix F.
A.4. LEGENDRE POLYNOMIALS: PN (X) 501

If
∞ ∞
f (x) ≡ xf (x) ≡ (A.38)
an Pn (x), bn Pn (x)
n=0 n=0
∞ ∞
2
df df
≡ ≡
a(1) Pn (x), a(2) Pn (x) (A.39)
dx n=0 n n
dx2 n=0

then the coef¬cients of the derived series are given in terms of the coef¬cients of f (x) itself
as

Multiplication by x:


n n+1
n ≥ 1 [b0 = 0] (A.40)
bn = an’1 + an+1 ,
2n ’ 1 2n + 3
First Derivative:


a(1) (A.41)
= (2n + 1) ap
n
p=n+1,p+n odd

Second Derivative:


{p(p + 1) ’ n(n + 1)} ap
a(2) (A.42)
= (n + 1/2)
n
p=n+2,p+n even
APPENDIX A. A BESTIARY OF BASIS FUNCTIONS
502

A.5 Gegenbauer Polynomials


Applications: Spherical geometry as the latitudinal factors for the spherical harmonics,
Gottlieb et al regularization of Gibbs™ Phenomenon

Alternative name: “Ultraspherical” polynomials

x ∈ [’1, 1]
Interval:



Weight Function:

m’1/2
ρ(x) = 1 ’ x2 (A.43)

Inner Product:

1
π 21’2m “(n + 2m)
(1 ’ x ) ≡ hm
(m) (m) 2 m’1/2
(A.44)
Cn Cn dx = n
2
n! (n + m) “(m)
’1

Expansion Coef¬cients of a Function f (x)



am Cj (x)
m
(A.45)
f (x) = j
j=0


1
1
≡m f (x) Cj (x) (1 ’ x2 )m’1/2 dx
am m
(A.46)
j
hj ’1

where hm is de¬ned by A.44 and where the Gegenbauer polynomials are not orthonormal,

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. 102
( 136 .)



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