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The method
Many orthogonal bases exist
In example:
Spherical harmonics,
, are a
) ,(
m
lY
series of functions defined on the surface
of a sphere that exhibit orthogonality:
ˆ
ˆ
m
l
ddθ i) ('
) (
mY mY
'
)ˆ(
)ˆ, ,(
) ,(
,
0
) (
) ,( ,
) , ,(
d m
lY Er
Er l
l
m
lY
l m
Erml
Er
ll' δ'
δ dφ
1
)
(cos
) ('
) (
φ
s n
2
,
'
,
d
mY mY
l
l
and therefore they can be used to expand
4
,
m
mm
2 1
,
'
,
l
l
any function defined on the surface of a
sphere in an infinite series:
l
d mY f
C
l
m
lY
l m
mlC
f
) (
) (
,
0
) ,(
,
) ,(
l
ml
,
,
,