Information Technology Reference
In-Depth Information
where ˛ is a characteristic scale. The abovementioned basis functions have an input
range from 1 to C1. The basis functions of Eq. (
12.10
) also satisfy orthogonality
condition, i.e.
R
C1
1
(12.11)
ˇ
m
.x;˛/ˇ
k
.x;˛/
dx
D ı
mk
;
where ı
mk
is the Kronecker delta symbol [
143
]. Any continuous function f.x/;x 2
R can be written as a weighted sum of the above orthogonal basis functions, i.e.
X
1
f.x/D
c
k
ˇ
k
.x;˛/
(12.12)
kD0
The expansion of f.x/ using Eq. (
12.12
) is a Gauss-Hermite series. It holds
that the Fourier transform of the basis function
k
.x/ of Eq. (
12.9
) satisfies the
relation [
143
]
k
.s/ D j
n
k
.s/
(12.13)
while for the basis functions ˇ
k
.x;˛/ of Eq. (
12.10
), it holds that the associated
Fourier transform is
B
k
.s;˛/ D j
n
ˇ
k
.s;˛
1
/
(12.14)
Therefore, it holds
f.x/D
P
kD0
c
k
ˇ
k
.x;˛/
F
F.s/D
P
kD0
c
k
j
n
ˇ
k
.s;˛
1
/
(12.15)
!
which means that the Fourier transform of Eq. (
12.12
) is the same as the initial
function , subject only to a change of scale.
12.4.2
Neural Networks Based on the Eigenstates
of the 2D Quantum Harmonic Oscillator
FNN with Hermite basis functions of two variables can be constructed by taking
products of the one-dimensional basis functions B
k
.x;˛/ [
143
]. Thus, setting x D
Œx
1
;x
2
T
one can define the following basis functions
1
˛
B
k
1
.x
1
;˛/B
k
2
.x
2
;˛/
B
k
1
;k
2
.x;˛/ D
(12.16)
