F.1 Introduction

The cardinal functions Cj (x) for a given type of interpolation (trigonometric, polynomial,

etc.) and for a set of interpolation points xi are de¬ned by the requirement that

(F.1)

Cj (xi ) = δij i, j = 1, . . . , N

where δij is the usual Kronecker delta symbol de¬ned by

1 i=j

δij ≡ [Kronecker delta] (F.2)

0 i=j

There is no universal terminology; other authors refer to the cardinal functions as the “car-

dinal basis”, “Lagrange basis”, or “the fundamental polynomials of interpolation”.

δk denotes the matrix of the k-th derivative at the interpolation points:

dk Cj

(δk )ij ≡ (F.3)

dxk x=xi

The tables use the auxiliary column vectors de¬ned by:

j = ±N j = 0 or N

2 2

cj ≡ pj ≡ (F.4)

&

otherwise otherwise

1 1

561

APPENDIX F. CARDINAL FUNCTIONS

562

F.2 General Fourier Series: Endpoint Grid

Taken from Gottlieb, Hussaini, and Orszag(1984).

Grid points:

πi

i = 0, . . . , 2N ’ 1 (F.5)

xi =

N

Cardinal Funcs.:

1

Cj (x) ≡ sin[N (x ’ xj )] cot[0.5(x ’ xj )] (F.6)

2N

N

1 1

≡ exp[ik(x ’ xj )] (F.7)

2N ck

k=’N

1st Derivative:

0 i=j

(δ1 )ij ≡ (F.8)

0.5(’1)i’j cot[0.5(xi ’ xj )] i=j

2d Derivative:

’(1 + 2N 2 )/6 i=j

(δ2 )ij ≡ (F.9)

0.5(’1)i’j+1 / sin2 [0.5(xi ’ xj )] i=j

F.3. FOURIER COSINE SERIES: ENDPOINT GRID 563

F.3 Fourier Cosine Series: Endpoint Grid

πi

[(N +1) degrees of freedom] (F.10)

xi = i = 0, . . . , N

N

±

sin(x) sin(N x)

(’1)j+1

pj N [cos(x) ’ cos(xj )]

Cj (x) ≡

cos

(F.11)

N

2 1

cos(mxj ) cos(mx)

N pj pm

m=0

where pj = 2 if j = 0 or N and pj = 1 if j = 1, . . . , N ’ 1.

1st Deriv.: (N + 1) — (N + 1) matrix

cos j = 0 or N

(δ1 )ij

dCj

≡ (δ1 )ij ≡

cos

(F.12)

dx otherwise

(δ1 )ij + (δ1 )i,2N ’j

x=xi

where δ1 is the ¬rst derivative matrix for a general Fourier series (F.8).

Alternative 1st Derivative:

±

i = 0 or i = N, all j

0

i = j; j = 1, . . . , N ’ 1

0.5 cot(xj )

(δ1 )ij ≡

cos

(F.13)

(’1)j+1 sin(xi ) cos(N xi )

i = j, 0, N

pj [cos(xi ) ’ cos(xj )]

2d Derivative:

δ 2 ≡ δ1 δ1

cos sin cos

(F.14)

sin

where δ1 is the ¬rst derivative matrix for a Fourier sine series, de¬ned in the next subsec-

tion.

APPENDIX F. CARDINAL FUNCTIONS

564

Quadrature formula: If f (x) = f (’x), then

N

π

f (x) dx ≈ (F.15)

wi f (xi )

0 i=0

where the xi are given by (F.10) and where the weights are

i = 1, 2, . . . , N ’ 1

π/N

wi ≡ (F.16)

π/(2N ) i = 0 or i = N

These weights are identical with those of the usual trapezoidal rule, but the accuracy for

this special case of a periodic, symmetric function is exponentially accurate. The quadrature is

exact if f (x) is a trigonometric cosine polynomial of degree at most N (where the number

of interpolation points is N + 1).

F.4. FOURIER SINE SERIES: ENDPOINT GRID 565

F.4 Fourier Sine Series: Endpoint Grid

πi

i = 1, . . . , N ’ 1 [(N -1) degrees of freedom] (F.17)

xi =

N

Note: by symmetry, a Fourier sine series must vanish at x = 0 & π, so there are only

(N ’ 1) degrees of freedom. We can de¬ne the derivative matrix as one of dimension

(N + 1) — (N + 1) for use in computing the even derivatives of a cosine series [by extending

(F.17) to include i = 0 and i = N ], but in applying collocation to compute a solution in the

form of a sine series, we should use only the interior points.

Cardinal Funcs:

(’1)j+1 sin(xj ) sin(N x)

Cj (x) ≡ j = 1, . . . , N ’ 1

sin

(F.18a)

N [cos(x) ’ cos(xj )]

N ’1

2

≡ j = 1, . . . , N ’ 1

sin

(F.18b)

Cj (x) sin(mxj ) sin(mx)

N m=1

1st Deriv.: (N + 1) — (N + 1) matrix

j = 0 or N

0

≡

sin

(F.19)

δ1 ij

(δ1 )ij ’ (δ1 )i,2N ’j otherwise

where δ1 is the ¬rst derivative matrix for a general Fourier series, de¬ned in (F.8), and

where cj is de¬ned by (F.4). An equivalent de¬nition of the sine derivative is

Alternative 1st Deriv:

±

j = 0 or N

0

’0.5 cot(xj ) i=j

≡

sin

(F.20)

δ1

ij

(’1)i+j+1 sin(xj )

otherwise

cos(xi ) ’ cos(xj )

2d Deriv:

sin cos sin

(F.21)

δ2 = δ1 δ1

APPENDIX F. CARDINAL FUNCTIONS

566

Exponentially Accurate Quadrature:

If f (x) = ’ f (’x), then

π

(F.22)

f (x) dx = wi f (xi )

0 i=1

where the xi are de¬ned by (F.17) and the weights are given by

N ’1

2 1

wi ≡ sin(mxi )[1 ’ cos(mπ)] (F.23)

N m

m=1

This quadrature formula is exact if f (x) is a trigonometric sine polynomial of degree (N ’1)

or less; note that (N ’ 1) is the actual number of quadrature abscissas.

F.5. COSINE CARDINAL FUNCTIONS: INTERIOR GRID 567

F.5 Cosine Cardinal Functions on the Interior (Rectangle Rule)

Grid

Previously unpublished.

Grid points:

(2i ’ 1) π

[N degrees of freedom] (F.24)

xi =

2N

cos(N x) sin(xj )

Cj (x) ≡ (’1)j+1

cos

(F.25)

N [cos(x) ’ cos(xj )]

N