xt=∑kak
ϕ
k
t
x
t
k
a
k
ϕ
k
t
(1)
How to find the coefficients
ak
a
k
? If the functions
ϕ
k
t
ϕ
k
t
form an
orthonormal set, meaning that
∫
ϕ
k
t
ϕ
m
tdt=δk-m=1ifk=m0ifk≠m
t
ϕ
k
t
ϕ
m
t
δ
k
m
1
k
m
0
k
m
(2)
then the coefficients
ak
a
k
can be found as follows.
xt=∑mam
ϕ
m
t
x
t
m
a
m
ϕ
m
t
(3)
xt
ϕ
k
k=∑mam
ϕ
m
t
ϕ
k
t
x
t
ϕ
k
k
m
a
m
ϕ
m
t
ϕ
k
t
(4)
∫xt
ϕ
k
kdt=∫∑mam
ϕ
m
t
ϕ
k
tdt
t
x
t
ϕ
k
k
t
m
a
m
ϕ
m
t
ϕ
k
t
(5)
∫xt
ϕ
k
kdt=∑mam∫
ϕ
m
t
ϕ
k
tdt
t
x
t
ϕ
k
k
m
a
m
t
ϕ
m
t
ϕ
k
t
(6)
∫xt
ϕ
k
kdt=∑mamδk-m
t
x
t
ϕ
k
k
m
a
m
δ
k
m
(7)
∫xt
ϕ
k
kdt=ak
t
x
t
ϕ
k
k
a
k
(8)
so we have the following equation for the coefficients:
ak=∫xt
ϕ
k
kdt
a
k
t
x
t
ϕ
k
k
(9)