My PhD thesis - Condensed Matter Theory - Imperial College London

APPENDIX D. RECONSTRUCTING A PROBABILITY DENSITY FUNCTION
D.2 Projection onto basis functions
The second approach to the problem proceeds by writing f(x) in terms of some
appropriate 1 basis:
∞∑
f(x) = c i χ i (x)
(D.13)
i=1
where {χ i } is a set of complete orthonormal functions 2 and
∫
c i = f(x)χ i (x) dx.
(D.14)
An approximation to c i may be obtained from the set of sampled points {x i }:
d i = 1 N
The expectation value of d i is
N∑
χ i (x j ).
(D.15)
j=1
〈
di
〉
=
〈
χi (x j ) 〉 x j
= c i . (D.16)
Using these approximate coefficients, an attempt at a reconstructed function is then
N
∑ χ
g(x) = d i χ i (x)
i=1
= 1 N
N
∑ χ
i=1
N∑
χ i (x j )χ i (x).
j=1
(D.17)
Only a finite number N χ of basis functions has been used. Comparison with equation
(D.1) shows that the technique discussed in the previous section is approximately a
special case of this more general method, with
N
∑ χ
w(x j , x) = χ i (x j )χ i (x).
i=1
(D.18)
1 The choice of basis is of course influenced by whatever prior knowledge of f is available: if f
has a certain symmetry, then only basis functions with the same symmetry need be considered.
Likewise, if f satisfies certain boundary conditions, the basis functions can be chosen to satisfy the
same constraints.
2 The basis functions in fact need only be linearly independent, not strictly orthogonal. However,
the analysis is much simpler in the case of an orthonormal basis.
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