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The method
Many orthogonal bases exist
In example:
Legendre polynomials, P
n
(x), are a series of functions that exhibit orthogonality
for -1≤x≤1:
and therefore they can be used to expand any
function defined in [-1,1] in an infinite series:
xnPnw xf
), (
)(
), ˆ ˆ( )
,(
4
1 2
)ˆ ˆ,
,(
nPE Erns
n
E Ers
dx xnPxf
nw
n
)(
1
)(
0
)ˆ ˆ( )ˆ ˆ(
1
)ˆ ˆ,
,(
2/1 )
,( ,
,
0
d
nP
E Ers
E Erns
n
1
1