# Earth Cubed

## Small Region Aprox. of Spherical Harmonics

The beauty of the method I orginaly proposed, is that if you take a small enough region of the earth the grid becomes rectangular. This means, that over a small region,  spherical harmonics, will look more like sine, and cosine functions.

If we use sine and cosine functions to approximate spherical harmonics over a small region the mathematics is greatly simplified and we can use standard FFTs, to compute the spectral components over this region. In fact little will be gained in this case by using the FFT instead of the DFT because we can use information about the spectral components, computed over one small region, to reduce the computation of the spectral components in an overlapping spectral region. This recurive comutation, will only be of linear complexity while the FFT is log-linear complexity.

Of course on a large scale we want to take fall advantage of the FFT like alrorithims to compute the spectral components, but when it comes to small scales we want to use interpolationg functions which capture the local rather then the global behavior and consequently we need a new DFT for each small overlapping region and therefore recursive algorithims are more efficient then FFTs.

Some consideration needs to be given as to weather, a Fourier Series is a good interpolating function for small regions of Navier stokes equations.  Aproximation, can be done with regards to another bais but the foier series, has a lot of advantages in terms of computational complxity. Even if another basis is used, it can be used in conjuction with the FFT by using one of the basis to interpolate the error given that truncated expansion in terms of the other basis.