PX275 - H3 - fourier transforms

considering a function defined over $x \in [- \infty, \infty]$ , but it is not periodic
eg: $f (x) = \exp (- x^{2})$
considering the limit where the interval of periodicity, $L \to \infty$

f (x) = \sum_{{k}} C_{k} \exp (i k x)

the above is the sum of plane waves with wave number, $k = 2 π n / L$
the separation between $k$ values:

δ k = \frac{2 π}{L}

as $L$ increases, $δ k$ gets smaller and smaller

lim_{L \to \infty} δ k \to 0

this forms a continuum of $k$
the forward transform:

C_{k} = \frac{1}{L} \int_{- L / 2}^{L / 2} f (x) e^{- i k x} d x

considering the limit:

F (f (x)) = \int_{- \infty}^{\infty} f (x) e^{- i k x} d x = \tilde{f} (k) (= lim_{L \to \infty} L C_{k})

this is defined to be the fourier transform of $f (x)$

\begin{aligned} f (x) & = \sum_{{k}} C_{k} e^{i k x} \\ = \sum_{{k}} δ k \frac{L}{2 π} C_{k} e^{i k x} \\ = \frac{1}{2 π} \sum_{{k}} L C_{k} e^{i k x} δ k \end{aligned}

as $L \to \infty$ , $δ k \to 0 \to d k$

lim_{L \to \infty} \frac{1}{2 π} \sum_{{k}} L C_{k} e^{i k x} δ k = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{f} (k) e^{i k x} d k = f (x)

this is defined as the inverse fourier transform

f (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{f} (k) e^{i k x} d k

fourier transforms map between 'real space', $x$ , and the corresponding reciprocal 'fourier space', $k$