The "normal" Pade Approximation is irrelevant. Could you point out why exactly this is superior or actually solves the issue? Any paper comparing these methods would also be interesting.
One way I handle the 'simple case that Fourier has trouble with' is I add the 'simple' bases to the catalogue of Fourier basis functions and then orthogonalise the resulting basis system.
Depending on specific 'simple' cases this can be messy but effective.
The same deal happens for decision trees. (Axis parallel) Trees have a hard time fitting linear functions, requiring many modes. One can use essentially the same solution.
BTW very informative commentary. The things that you bring up really needed to be brought out.
One way I handle the 'simple case that Fourier has trouble with' is I add the 'simple' bases to the catalogue of Fourier basis functions and then orthogonalise the resulting basis system.
Depending on specific 'simple' cases this can be messy but effective.
The same deal happens for decision trees. (Axis parallel) Trees have a hard time fitting linear functions, requiring many modes. One can use essentially the same solution.
BTW very informative commentary. The things that you bring up really needed to be brought out.