1. ## fourrier transform proof

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

2. Originally Posted by allah's_slave
is there a proof 4 fourrier's theory;that any signal is composed of infinite number of sinusoids?

For a continuous absolutely integrable function $\displaystyle f(x)$ with absolutly integrable Fourier transform $\displaystyle [\mathcal{F}f](\xi)=F(\xi)$ the result follows from:

$\displaystyle f(0)=\int_{-\infty}^{\infty} F(\xi)~d\xi \ \ \ \ \ \ \ ...(1)$

by the translation theorem.

To prove $\displaystyle (1)$ you need a sequence of well behaved functions $\displaystyle u_n(x)$ with known Fourier transforms $\displaystyle U_n(\xi)$, which as $\displaystyle n \to \infty$ approximates the behaviour of a $\displaystyle \delta$ 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 $\displaystyle (1)$ you consider:

$\displaystyle \lim_{n \to \infty} \int f(x)u_n(x) e^{i 2 \pi \xi x}dx$$\displaystyle \ = \lim_{n \to \infty }(F*U_n)(\xi)$

By construction the limit on the left hand side is $\displaystyle f(0)$, and on the right hand side you can exchange the limit and integral and so find it goes to right hand side of $\displaystyle (1)$.

(Of course this is much more direct it you run amok with generalised functions or distributions, when you use $\displaystyle \delta(x)$ instead of $\displaystyle \{ u_n(x)\}$, and $\displaystyle 1$ instead of $\displaystyle \{U_n(\xi)\}$)

(And also this is modulo the odd factor depending on how the FT has been defined)

RonL