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⎛
(
)
2
⎞
N
−
1
N
−
1
(
)
i
π
⎜
j
NN
−
−
1
⎟
π
(
)
⎜
2
⎟
()
()
⎝
⎠
U
t
=
e
fft
ν
t
,
j
=
1,2,
…
,
N
J
j
2
N
where
fft
(
v
(
t
)) stands for the output of any standard FFT algorithm applied
to the vector
v
(
t
).
Instead of approximating at the scaled points
aκ
j
, it is better in some
cases to approximate just at the nodes
κ
j
. This can be done by scaling the
function
u
(
x, t
) to yield
()
u x t
,
=
u ax t
(
, )
and
∞
(
)
(
)
∫
ik x
⋅
Uk at
/
,
=
a e uxtdx
,
−∞
Since the XFT gives a scaled approximation, we have that
(
)
()
⎡
()
⎤
Ut
κ
,
≅
Ut
=
xft
ut
lim
,
j
=
1,2,
…
,
N
,
⎣
⎦
j
j
1
x
→∞
j
()
where
stands for the output of the one-dimensional XFT ap-
xft
⎡
u t
⎤
⎣
⎦
1
()
plied to
u
. For the inverse transform, we have a similar result
u
j
(
t
)=
aixft
1
[
U
(
t
)]
j
j
=1,2,
…,N ,
(6)
()
⎡
⎤
ixft
U t
where now
stands for the output of the inverse XFT applied
⎣
⎦
1
U
.
This algorithm is based on the approximation of a square-integrable
function by a linear combination of discrete Hermite functions [11]. Be-
cause of this, it gives accurate results for the computation of the Fourier
transform of rapidly decreasing functions evaluated at the nodes
κ
j
.
The generalization of these results to more dimensions can be done
straightforward, and the approximated and scaled version of Eq. (4) is
found to be the vector of
N
n
components
()
to
2
()
at
a
κ
U
0 (
=
e
U t
( ),
q
q
ordered according to the following equation:
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