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From: Joubert on 27 Jul 2010 19:31
I have notes written by another guy some parts of which are totally
obscure for many different reasons. I was just hoping someone had seen
something similar and would recall the book they read it on. Although
Gel'fand had other useful calculations I found it, luckily, in "Harmonic
analysis in the phase space".
From: Ray Vickson on 28 Jul 2010 00:46
On Jul 27, 4:31 pm, Joubert <waterhemlock8...(a)gmail.com> wrote:
> I have notes written by another guy some parts of which are totally
> obscure for many different reasons. I was just hoping someone had seen
> something similar and would recall the book they read it on. Although
> Gel'fand had other useful calculations I found it, luckily, in "Harmonic
> analysis in the phase space".

Using Christopher Henrich's suggestion, I get "almost" your desired
result. Let f(x) = exp(i*Pi*<x,T,x> - e*Pi*<x,x>), where e ("epsilon")
> 0 is a small parameter, taken to zero at the end. (Here, <a,b> =
inner product of vectors a and b.) The FT of f is g(w) = int(f(x)*exp(-
i*<w,x>) d^n x, x in R^n) for w in R^n. Since T is invertible and real-
symmetric, there exists an orthogonal matrix Q (i.e., Q^T * Q = I_n)
such that T = Q^T * A * Q, where A = diag(a_1,a_2,...,a_n) and all a_j
<> 0 by invertibility. Letting t = Q*w and y = Q*x we have g =
int(exp(-i*<t,y> + i*Pi*<y,A,y> - e*Pi*<y,y>) d^n y, y in R^n); this
is true because d^n x = |det(Q)| * d^n y and |det(Q)| = 1 [because 1 =
det(I_n) = det(Q)^2], and because <x,Tx> = <x,Q^T,A Qx> = <Qx,A Qx> =
<y,A,y> and <x,x> = <x,Q^T Q x> = <Qx,Qx> = <y,y>. We have that g is a
product of integrals of the form g_j = int(exp(-i*t_j*y_j +
i*Pi*a_j*y^j^2 - Pi*e*y_j^2) dy, y in R). Taking the limit e --> 0
after doing the integrals we have the following. For a_j > 0 we have
g_j = exp(-i*3*Pi/4)*exp(-(i/(4Pi))*t_j^2/a_j)/sqrt(a_j). For a_j = -
b_j (where b_j > 0) we have g_j = exp(-i*Pi/4)*exp(-(i/(4Pi))*t_j^2 /
a_j)/sqrt(b_j) [where we have written 1/a_j in the exp() and b_j in
the sqrt()]. The whole Fourier transform of f is the product of the
g_j, and noting that exp(-i*3*Pi/4) = exp(-i*Pi)*exp(i*Pi/4), we get g
= (-1)^p * exp(i*Pi*s/4)*U/sqrt(|a_1*a_2*...*a_n|), where U = exp[-(i/
(4Pi))*sum(t_j^2/a_j)], p = number of positive a_j and s = number
positive a_j - number negative a_j. Note that sqrt(..) = sqrt(|
det(A)|) = sqrt(|det(T)|) and the sum in the exponent for U is
<t,A^(-1)t> = <Qw,A^(-1) Qw> = <w,Q^(-1) A^(-1) Q w> = <w,T^(-1)w>.
So, aside from the factor (-1)^p, having sqrt(|set(T)|) instead of
sqrt(det(T)) and some Pi factors in the exponent, this is the result
you want. The Pi factors would go away with a slightly different
normalization of the FT, so the only remaining question is why I get
the extra factor (-1)^p and have |det(T)| instead of det(T). Perhaps
if you go over the computations again, carefully, you will get rid of
this discrepency.

R.G. Vickson
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