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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

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