interchanging summation and integration:
w0, A ʌ V ∏ (-1)n lθsτx + θvτy]2n
W (θaτ ,θυτ ) = > y J J --------(2n)!--------K (x,y) dxdy.
n=o_1 _x
Next, in the grid, θsτ fcι = ∆1sτ,k1 = 2^1, θυτ k? = ^2vτ,k2 = 2J^2 , then, on the rectangle of integration:
I^ST,k1 X + θVτ,k2 У\ — θSτ,k1 lXl + θVτ,k2 |y| — θSτ,k1 + θυτ,k2 < 4^.
So,
ʃ ʃ [θsτx + θvτy]2n K (x, y) dxdy
_1 _1
1 1
— ʃ ʃ ∣θsτx + θυτy∣2n K (x,y) dxdy < (4π)2n.
_1 _1
(2n)! swamps
(4π)2n at a fast rate. Of course, there will be additional attenuation due to the kernel function itself.
Therefore:
Æ
W (θsτ ,θvτ ) = 1 + V
(-1)n
(2n)!
1 1
// ^sτ
_1 _1
x + θυτy]2n K (x, y) dxdy,
even for a moderate choice of N. Still, truncation at a very low N, may result in a negative value of W.
Closed-form expressions for the integrals are straightforward to derive and are not presented here to save
space. Some experimentation reveals that very accurate results for W are obtained for N = 8. This choice
also guarantees strict positivity of W for all θsτ, θυτ ∈ [0, 2π].
21
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