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I'm solving an ODE, where currently the dependent variables are the (time dependent) spatial Fourier coefficients. It turns out that the phenomena I'm interested in describing is spatially localized, and it takes an impractical number of Fourier coefficients to fully describe it. Therefore, I'm interested in looking at different basis to use, instead of the canonical trig functions $e^{ikx}$ (for $k\in \mathbb{Z}$).

However, for this model I need these basis functions to have certain properties. In particular, I need these functions to be differentiable, as well as periodic. Also, I need the functions to have relatively simple Hilbert Transforms, since this is exploited in solving for the governing equations (and is one reason why the trigonometric basis was desirable). Finally, the basis should converge more rapidly than the Fourier series for spatially compact scenarios. Hence, I would like to reduce the number of dependent variables that I'm solving for in my ODE.

Does anyone know of any functions fitting the bill? Any tips/references are appreciated.

Nick

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    $\begingroup$ Why does the basis need to be periodic if the function being approximated is localized? If this were not a consideration I would suggest using smooth wavelets. For an understanding of how the Hilbert transforms acts on wavelets (generally it gives back another wavelet), see: bigwww.epfl.ch/chaudhury/HilbertTransform_Wavelets.pdf $\endgroup$ – Nick Alger Jul 13 '14 at 18:18

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