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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:
dx xnPxf
nw
xnP
n
nw xf
)
(
1
1
)(
), (
0
)
(
−
=
=
=
)ˆ ˆ( )ˆ
ˆ(
1
1
)ˆ ˆ,
,(
2
/1 )
,
( ,
), ˆ ˆ( )
,( ,
0 4
1 2
)ˆ ˆ,
,(
−
→
=→
→
=
+
=
→
d
nP
E Ers
E Erns
nPE E
rns
n
n
E Ers
and therefore they can be used to expand any
function defined in [-1,1] in an infinite series: