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In-Depth Information
a
b
Hermite basis functions
2D Hermite basis function
1
0.8
1
0.6
0.4
0.5
0.2
0
0
−0.2
−0.5
−0.4
−1
10
−0.6
5
10
−0.8
0
5
0
−5
−5
−1
−10
−10
−8
−6
−4
−2
0
2
4
6
8
10
y
x
−10
x
Fig. 12.2
(
a
) First five one-dimensional Hermite basis functions (
b
) 2D Neural Network based on
the QHO eigenstates: basis function B
1;2
.x;˛/
According to Eq. (
12.8
) the first five Hermite polynomials are:
H
0
.x/ D 1 for D 1;a
1
D a
2
D 0; H
1
.x/ D 2x for D 3;a
0
D a
3
D 0
H
2
.x/ D 4x
2
2 for D 5;a
1
;a
4
D 0; H
3
.x/ D 8x
3
12x for D 7;a
0
;a
5
D0
H
4
.x/ D 16x
4
48x
2
C 12 for D 9;a
1
;a
6
D 0
It is known that Hermite polynomials are orthogonal [
186
]. Other polynomials
with the property of orthogonality are: Legendre polynomials, Chebychev polyno-
mials, and Laguerre polynomials [
215
,
217
].
12.4
Neural Networks Based on the QHO Eigenstates
12.4.1
The Gauss-Hermite Series Expansion
The following normalized basis functions can now be defined [
143
]:
x
2
2
;kD 0;1;2;
k
.x/ D Œ2
k
2
kŠ
1
2
H
k
.x/e
(12.9)
where H
k
.x/ is the associated Hermite polynomial. To succeed multi-resolution
analysis Hermite basis functions of Eq. (
12.9
) are multiplied with the scale coeffi-
cient ˛. Thus the following basis functions are derived
ˇ
k
.x;˛/ D ˛
2
k
.˛
1
x/
(12.10)
