I'm looking for the Fourier transformation of the (constant) uniform B-Spline $N_0(x) = \begin{cases}1 & 0 \leqslant x < 1 \\ 0 & otherwise \end{cases}$
If $N_0(x)$ would also attain value $1$ when $x=1$ (i.e. the $<$ would be $\leqslant$), it would just be a shifted version of the rectangular function R(x), which has (if I'm not mistaken) the following Fourier transform ($T=1$ and shifted $1/2$ to the right):
$\hat{R}(\omega) = e^{-\tfrac{i\omega}{2}}\, \frac{\sin(\omega/2)}{\omega/2}$
However, I'm not sure if this is helpful in finding $\hat{N_0}(\omega)$, the Fourier transform of $N_0(x)$...
Once I know $\hat{N_0}(\omega)$, it is easy to find the Fourier transforms of higher order uniform B-Splines, since they are (or can be) defined using convolution. Of course, the Fourier transform of a convolution is just a multiplication of the two Fourier transformed functions, therefore:
$\hat{N_k}(\omega) = \left( \hat{N_0}(\omega) \right)^{k+1}$