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Convergence with number of sample points--theory

What is the minimum required for a given accuracy in (G.2)? Writing as a complex Fourier series on the periodic interval ,

 (G.4)

it is clear that the desired integral is just . The approximation (G.2) instead gives the sum . The notation implies the sum of and both with arbitrary phase factors. This is an example of the Nyquist sampling theorem[161]: is completely represented up to spatial frequencies , but beyond this the spectrum is folded back into this range. In particular the frequencies for integer get folded back to zero-frequency, corrupting the estimate of . Thus one should choose sample spacing , where is the bandwidth of . That is, is assumed to have negligible components beyond wavenumber .

The functions and are wavefunctions (Helmholtz equation solutions existing in the plane), measured along a closed curve, which I first take as smooth and of large curvature radius. So to a first approximation and are bandwidth-limited to , where the free-space wavenumber is . Therefore is bandwidth-limited to . This would suggest that just over 2 samples per free-space wavelength are needed to achieve high accuracy, and that convergence is exponential beyond this.

Two facts heavily modify this simple semiclassical picture: 1) there exist evanescent wave components in eigenstates or basis functions (for instance, evanescent basis functions can oscillate at up to , see Section 6.3.3), and 2) the boundary may not be infinitely differentiable (required for exponential convergence), rather corners (and `kinks') may introduce discontinuities into the function which give power-law Fourier tails (and power-law convergence).

Next: Convergence in practice Up: Appendix G: Numerical evaluation Previous: Appendix G: Numerical evaluation
Alex Barnett 2001-10-03