What exactly is Fourier Transformation? For functions on the Schwartz Space $S(\Bbb R^n)$, we may define,
$$ \hat{u}(\xi) := \int e^{-ix\xi} u(x) \, dx $$ This formula seems to come out of no where for me.
It satisfies so many amazing properties that seems nonintuitive, to name a few:
(1) $$ ||f||_2 =||\hat{f}||_2$$
(2) $$ \hat{\hat{u}} = u^{-}$$
Is there a general/intuitive interpretation for how (i) One comes up with the formula, and (ii) why we should expect results as (1) and (2)?
On
Fourier originally came up with the Fourier transform, but did so using $\sin$ and $\cos$ functions. The exponential series and transforms weren't used at that time. Fourier used a limit of the Fourier series as the period grew longer and longer in order to come up with integral expansions. He argued that the sums turned to integrals as the period grew without bound. His arguments were not rigorous, but they resulted in correct expressions. The exponential series on $[-L,L]$ is $$ f(x) = \sum_{n=-\infty}^{\infty}\frac{1}{2\pi L}\int_{-L\pi}^{L\pi}f(y)e^{-iny/L}dy\cdot e^{inx/L} $$
You can imagine--though probably not correctly justify--that the limit as $L\rightarrow\infty$ looks like a Riemann sum for $$ f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}f(y)e^{-isy}dy\cdot e^{isx}ds $$ I've never seen a correct argument where this is made rigorous; but it is good motivation for the continuous case, and the final result is correct.
The discrete expansion of $f$ is a series $$ \sum_{n=-\infty}^{\infty}\langle f,e^{-iny/L}\rangle e^{inx/L} $$ where $\langle f,e^{-iny/L}\rangle = \frac{1}{2\pi L}\int_{-L\pi}^{L\pi}f(y)e^{-iny/L}dy$ is the amplitude of $e^{inx/L}$. The limit as $L\rightarrow\infty$ gives way to an integral, which may be interpreted as a "continuous" sum $\int$ given by $$ f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \langle f,e^{isy}\rangle e^{isx}ds, $$
where $ \langle f,e^{isy}\rangle = \int_{-\infty}^{\infty}f(y)e^{-isy}ds. $
The discrete Parseval relation $$ \|f\|^2 = \frac{1}{2\pi}\sum_{n=-\infty}^{\infty} |\langle f,e^{iny/L}\rangle|^2 $$ becomes an integral relation $$ \|f\|^2 = \frac{1}{2\pi}\int_{-\infty}^{\infty} |\langle f,e^{ins}\rangle|^2 ds $$ I think it's good to see this analogy between the discrete and continuous. The intuition is great, but it is not a substitute for a rigorous development which, unfortunately, cannot follow this line of argument in any simple way.
The basic idea was to decompose a function into sine and cosine waves with varying amplitudes. Complex numbers make the math quite a bit neater, but the basic idea is still the same even though technically we are now decomposing the function into "complex exponentials". The equation then comes from the fact that you can represent sinusoids with complex exponentials.
As for the properties, doing the math reveals the properties. But at least for the second property, Fourier Transform and its inverse are not the same so I wonder if that even holds.