This is a "voodoo" development

Fiω=∫−∞∞fte−(iωt)d
t
=〈ft,eiωt〉
F
ω
t
f
t
ω
t
f
t
ω
t

(1)
in Hilbert Space

L2R
L
2
, which is
like a generalized CTFS coefficient with "basis"

eiωt
ω
t
, given that

ω
ω is real.

All
ei
ω
1
t
ω
1
t
and
ei
ω
2
t
ω
2
t
are *orthogonal*
ω
1
≠
ω
2
ω
1
ω
2
, in other words:
〈ei
ω
1
t,ei
ω
2
t〉=0
ω
1
t
ω
2
t
0
unless
ω
1
=
ω
2
ω
1
ω
2
.

eiωt
ω
t
has infinite energy
〈eiωt,eiωt〉=∞
ω
t
ω
t
.

〈ei
ω
1
t,ei
ω
2
t〉=2πδ(
ω
1
−
ω
2
)
ω
1
t
ω
2
t
2
δ
ω
1
ω
2
.

If we carry on undeterred then we should be able to resynthesize
ft
f
t
from

ft=∑ω∈RFiωeiωt
f
t
ω
ω
R
F
ω
ω
t

(2)
where the sum becomes an integral:

ft=12π∫−∞∞Fiωeiωtd
ω
f
t
1
2
ω
F
ω
ω
t

(3)
ft=12π〈(Fiω,e−(iωt))〉
f
t
1
2
F
ω
ω
t
where
e−(iωt)
ω
t
is the inner product over
ω
ω
(why we need the
12π
1
2
).
Fiω=〈ft,eiωt〉
F
ω
f
t
ω
t