Multi-dimensional Fourier transforms in the helical coordinate system

Conclusion

We have explicitly found the relationship between multi-dimensional FFT's and 1-D FFT's on a helix, linking the wavenumber vector, ${\bf k}$, in a multi-dimensional system with the wavenumber of a helical 1-D FFT, $k_h$. Specifically, the Fourier dual of helical boundary conditions is helical boundary conditions but with axes transposed. We have illustrated the concepts with an example of multi-dimensional multiple prediction.