(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:

iπ

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,