8. Appendix
8.A. Kernel Function
Let the kernel function be:
k (x,y) =
_______________(ι-M)2(ι-H)2_______________
≈2y2 + ≈2(1-∣y∣)2 + (1-∣≈∣)2y2 + (1-∣≈∣)2(1-∣y∖)2 ’
0,
if both ∣x∣ ≤ 1 and ∣y∣ ≤ 1 .
elsewhere.
K (x,y) has the following desirable properties. First, it is symmetric and nonnegative.
Second, it reaches a max of 1 if both x = 0 and у = 0. Since x = ɑʌɪ21 , у = ɑʌ^22 this implies a weight
of 1 in case (ζ 1, ζ2) coincides with one interpolation point.
Third, since the grid of (ζ 1, ζ2) is equispaced, K (x, у) will assign positive weights to at most 4 interpolation
points closest to (ζ 1,ζ2).
Fourth, it can be verified that:
OO OO
ʃ ʃ K (x,y)
-∞ -∞
dxdy =
ff K (x,y)
dxdy = 1,
-1 -1
which establishes that K (x, y) is a valid kernel function. See Figure 5 for a graphical representation.
8.B. Weighting Function
½ Ъ1
(eST χ+θυτ y) K (x,y) dxdy. With the discussed choice of the kernel
By construction, W (θsτ ,θυτ ) = ʃ ʃ e-t
02 ɑl
function, this specializes as:
1 1
W (θsτ ,θvτ )
ʃ ʃe-1^ x+θ-τ y)K (x,y) dxdy =
-1 -1
1 1
ʃ ʃ cos(θ5τx ÷ θvτy)K (x,y) dxdy
-1 -1
i/ ∕sto(..r
-1 -1
x ÷ θvτy)K (x, y) dxdy
1 1
∕7"*θ-x ÷ ,„y)K (x,y) dxdy.
-1 -1
W (θsτ, θυτ) may be evaluated in a number of ways. A direct approach is to approximate it by numerical
integration.
A better way is to exploit the properties of the cosine function. Taylor expanding cos(θsτx ÷ θυτy) and
20
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