My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX D. RECONSTRUCTING A PROBABILITY DENSITY FUNCTION
However, there are some weight functions which cannot be reproduced as long as N
is finite.
The expected value of the reconstructed function is
〈 〉 ∑〈 g(x) χi (x j ) 〉 x j
χ i (x)
{x j } = N χ
i=1
N
∑ χ
= c i χ i (x).
i=1
(D.19)
This is a good approximation if c i = 0 for i > N χ .
Finally, the expected square deviation from the original function is
〈 (
g(x) − f(x)) 2
〉
{x j }
= 1
N 2
N
∑ χ
N N∑ ∑ χ
i=1 j=1 k=1 l=1
− 2f(x) 〈 g(x) 〉 {x i } + f(x)2
N∑
χ i (x)χ k (x) 〈 χ i (x j )χ k (x l ) 〉 x j ,x l
= 1 N
N
∑ χ N χ
i=1
+ N − 1
N
∑
χ i (x)χ k (x) 〈 χ i (x j )χ k (x j ) 〉 x j
k=1
N
∑ χ N χ
i=1
∑
χ i (x)χ k (x) 〈 χ i (x j ) 〉 〈
χk
x j
(x j ) 〉 x j
k=1
(D.20)
− 2f(x) 〈 g(x) 〉 + {x i } f(x)2
( ) 〈g(x) 〉
2
=
− f(x) {x i
+ 1 N χ N
∑ ∑ χ
χ
} i (x)χ k (x)
N
i=1 k=1
( 〈χi
× (x j )χ k (x j ) 〉 x j
− 〈 χ i (x j ) 〉 〈
χk
x j
(x j ) 〉 )
x j
.
This is analogous to equation (D.6). The first term is large when an insufficient
number of basis functions are used (compare equation (D.19)). However, the second
term becomes large when too many functions are used. To see this, consider what
happens as N χ → ∞. The second term becomes
∫
1
N
f(y)
{
∞∑
∑ ∞ ∫
χ i (x)χ i (y) χ k (x)χ k (y) −
i=1
= 1 N
∫
k=1
f(z)
( ∫
f(y)δ(x − y) δ(x − y) −
}
∞∑
χ k (x)χ k (z) dz dy
k=1
)
f(z)δ(x − z) dz dy,
(D.21)
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