available.
As noted by Press et al. (1992, p. 578), this approximation is apt to be imprecise. The sources of the
innacuracy are the possible error of truncation (if the function is not negligibly small at the cutoff boundary),
and the periodicity of the integrand function.
The first criticism does not apply as the property of vanishing at infinity has been established for the
integrand function. The second criticism does apply, however. Numerical integration of oscillatory functions
is a painstaking and often unsuccessful endeavor.
One possible way to mitigate the problem is to do kernel smoothing. I extend Press et al. solution to the
bivariate case. Interpolate Φ (C 1,C2) as follows:
φ(cι,<,> ≡ y y*j,λк(δς½.,δs½) + (и)
J2=O Ji=O ' 2
У φ. . K . (ζ 1 ~ ζJi ζ2 ~ ζJ2 )
ʌ ψji,j2Kji,j2 I A1 , д2 I ,
Ji ,J2 ∈{endpoi∏ts}
where K (ɑʌɪ^ , ^2δ^32 ) is the kernel function and Kjij2 (ɑʌɪ^ , ^2-2a-) is the difference between the
true kernel function at endpoints and K (∙, ∙).
Since the cutoff points are sufficiently large in absolute value, Φjijj2 for j1 ,j2 ∈ {endpoints} is negligibly
small. So, the second summation in (11) may be ignored.
b2 bi
Now, apply ʃ ʃ e-^i5τ+^2υτ>dζ 1 dζ2 to both sides of (11):
a2 ai
J Je-4<,τ +<2υτ >φ(C t,ζ 2) d( ld( 2 =
a2 ai
Г Γ , , Д1 ,Al /C, -C- C^ — C- ∖
11 e-l(CiST+ζ2υτ > y y φ ,J2 K ( δ, δ ) dC 1dC 2 =
α2 ai j2=0 ji=0 41 27
ʌt ʌ'1 b'r br CC- CC- C ∖
= У У /7'sτ+ζ2υτ>K (" δ', ⅛y2) dC 1dC2 Φji,j2∙
j2=0 ji=0 [a2 ai 41 2 7
Change variables as ^ɪʌ^ɪ = x, ^2-f2 = У :
'2 f (sτ,vτ) = Δ1Δ2W (θsτ ,θυτ) y y e-l(sTζ- +υτζ-)φj1,j2,
where W (θsτ ,θυτ ) =
Ь2 bi Z Z
ʃ ʃe-l(e^τx+θ^τy)K (x,y) dxdy
a2 ai
, θ sτ
= Δ1sτ, θυτ = Δ2Vτ. An important fact
10
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