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One-dimensional Fourier Transformation is defined as follows:
::{EQUATION()}
\label{Fourier1}
H(k) = \mathcal{F}\left( h(x) \right) = \int \limits_{-\infty}^{\infty} h(x) \mathrm{e}^{-\mathrm{i}2\pi kx} \mathrm{d}x
{EQUATION}(1)::
We mention here that the domain of this transformation is the set of functions whose absolute value has a finite integral over {EQUATION()}$\mathbb{R}${EQUATION} but hereinafter we will always suppose that the picked function is element of this set as in practice we will deal with finite functions limited in a finite time interval, and therefore their absolute value can always be integrated.
The inverse Fourier transformation is very similar to the transformation itself, they only differ in a complex conjugate:
::{EQUATION()}
\label{Fourier2}
h(x) = \mathcal{F}^{-1}\left( H(k) \right) = \int \limits_{-\infty}^{\infty} H(k) \mathrm{e}^{\mathrm{i}2\pi kx} \mathrm{d}k
{EQUATION}(2)::
The frequently used properties of the Fourier Transform are the followings:
1) Linearity:
::{EQUATION()}
\label{Fourier3}
\mathcal{F}\left( \alpha g(x) + \beta h(x) \right) = \alpha \mathcal{F} \left( g(x) \right) + \beta \mathcal{F} \left( h(x) \right)
{EQUATION}(3)::
for every complex numbers {EQUATION()}$\alpha${EQUATION} and {EQUATION()}$\beta${EQUATION}. Specifically, a constant phase multiplication is preserved during the Fourier transform.
2) Shifting:
::{EQUATION()}
\label{Fourier4}
\mathcal{F}\left( h(x-x_0) \right) = \mathcal{F} \left( h(x) \right) \mathrm{e}^{-\mathrm{i}2 \pi k x_0} = H(k) \mathrm{e}^{-\mathrm{i}2 \pi k x_0}
{EQUATION}(4)::
::{EQUATION()}
\label{Fourier5}
\mathcal{F}^{-1}\left( H(k-k_0) \right) = \mathcal{F}^{-1} \left( H(k) \right) \mathrm{e}^{\mathrm{i}2 \pi k_0 x} = h(x) \mathrm{e}^{\mathrm{i}2 \pi k_0 x}
{EQUATION}(5)::
For every real {EQUATION()}$k_0${EQUATION} and {EQUATION()}$x_0${EQUATION}. This means that a linear phase ramp in the Fourier space ({EQUATION()}$k${EQUATION}-space) results a spatial shift in the image space and vice versa. The shift is proportional to the steepness of the phase ramp.
3) Convolution:
::{EQUATION()}
\label{Fourier6}
\mathcal{F}\left( g(x) \Conv h(x) \right) = \mathcal{F} \left( g(x) \right) \mathcal{F} \left(h(x) \right) = G(k)H(k)
{EQUATION}(6)::
::{EQUATION()}
\label{Fourier7}
\mathcal{F}^{-1}\left( G(k)H(k) \right) = \mathcal{F}^{-1} \left( G(k) \right) \Conv \mathcal{F}^{-1} \left( H(k) \right) = g(x) \Conv h(x)
{EQUATION}(7)::
::{EQUATION()}
\label{Fourier8}
\mathcal{F}\left( g(x) h(x) \right) = \mathcal{F} \left( g(x) \right) \Conv \mathcal{F} \left(h(x) \right) = G(k) \Conv H(k)
{EQUATION}(8)::
::{EQUATION()}
\label{Fourier9}
\mathcal{F}^{-1}\left( G(k) \Conv H(k) \right) = \mathcal{F}^{-1} \left( G(k) \right) \mathcal{F}^{-1} \left( H(k) \right) = g(x) h(x)
{EQUATION}(9)::
In other words, a multiplication in the {EQUATION()}$k${EQUATION}-space results as a convolution in the image space with the inverse Fourier transformed version of the multiplying function. In the other direction, a convolution in the {EQUATION()}$k${EQUATION}-space means a multiplication in the image space.
4) Symmetry. If {EQUATION()}$h(x)${EQUATION} is a real function then:
::{EQUATION()}
\label{Fourier10}
H(-k) = \overline{H(k)}
{EQUATION}(10)::
This symmetry is of utmost importance in the so-called partial Fourier imaging where the effective spin density is assumed to be real, and therefore the {EQUATION()}$k${EQUATION}-space data is redundant as negative {EQUATION()}$k${EQUATION}-vector values can be replaced by the complex conjugate of the positive {EQUATION()}$k${EQUATION}-vector values.
The extension of the upper properties to higher dimensions can be done quite intuitively. The variables are then vectorial denoted by {EQUATION()}$\mathbf{r}${EQUATION} and {EQUATION()}$\mathbf{k}${EQUATION}, and the Fourier Transform (for example in 3D) is defined by the following with {EQUATION()}$\mathbf{kr}${EQUATION} equals to their dot product:
::{EQUATION()}
\label{Fourier10}
H(\mathbf{k}) = \int h(\mathbf{r}) \mathrm{e}^{-\mathrm{i}2\pi \mathbf{kr}} \mathrm{d^3r}
{EQUATION}(11)::