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The method
Many orthogonal bases exist
In example:
➢
Spherical harmonics,
, are a
series of functions defined on the surface
of a sphere that exhibit orthogonality:
and therefore they can be used to expand
any function defined on the surface of a
sphere in an infinite series:
) ,(
m
l
Y
=
=
−=
=
d m
l
Y f
mlC
l
m
l
Y
l
l m
mlC
f
) ,
(
) ,(
,
,
0
) ,(
,
) ,(
m: 0 1 2 3 …
0
1
l:
2
3
…
Various types of spherical harmonics
are available.
A particular set, of order l=N
(orthogonal
basis)
+ the
corresponding weights of a function
are called a
quadrature set of
order N
.
ll' δ mm'
δ dφ
2
1
1
)
(cos
) ,('
'
) ,(
dφ dθ sin
2
)
,('
'
) ,(
=
−
=
d
m
l
Y m
l
Y
m
l
Y m
l
Y