is there a proof 4 fourrier's theory;that any signal is composed of infinite number of sinusoids?
another question : what is the reason that made fourrier think of signals in that way? i mean , of course he faced some problem or had a certain idea before he put his theory
Originally Posted by allah's_slave
For a continuous absolutely integrable functionwith absolutly integrable Fourier transform
the result follows from:
by the translation theorem.
To proveyou need a sequence of well behaved functions
with known Fourier transforms
, which as
approximates the behaviour of a
functional.
A suitable sequence can be constructed from Gaussians with decreasing spread parameters (who's FTs form a sequence of Gaussians with increasing spread parameters).
Then to prove
u_n(x) e^{i 2 \pi \xi x}dx)
By construction the limit on the left hand side is, and on the right hand side you can exchange the limit and integral and so find it goes to right hand side of
(Of course this is much more direct it you run amok with generalised functions or distributions, when you useinstead of
, and
instead of
)
(And also this is modulo the odd factor depending on how the FT has been defined)
RonL
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