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Assuming that the range of the variable x is limited in the interval
from N to N, the discrete Fourier basis (Fast Fourier
Transform ) employs a set of orthonormal periodic functions
 
(20) 
where the discrete frequency index k also ranges, according to the
Nyquist sampling criterion, from N to N. The interpolation
function is computed from formula (13) to be
 

 
 (21) 
An interpolation function, equivalent to (21), has been
found by Muir Lin et al. (1993); Popovici et al. (1993). It can be
considered as a tapered version of the sinc interpolator
(19) with the smooth tapering function
Unlike triangletapered sinc interpolator, Muir's interpolator
(21) satisfies not only the obvious property
(2), but also properties (3) and
(17), where the interpolation function W (x,n)
should be set to zero for x outside the range from n  N to
n+N. The form of this function is shown in Figure
1.
masinc
Figure 1 The left plots show the sinc interpolation
function. Note the slow decay in x. The middle shows the
effective tapering function of Muir's interpolation; the right is
Muir's interpolator. The top is for N=2 (5point interpolation);
the bottom, N=6 (13point interpolation).
The development of the mathematical wavelet theory Daubechies (1992) has
opened the door to a whole universe of orthonormal function bases,
different from the Fourier basis. The wavelet theory should find many
useful applications in geophysical data interpolation, but exploring
this interesting opportunity goes beyond the scope of this paper.
The next section carries the analysis to the continuum and compares
the mathematical interpolation theory with seismic imaging
operators.
Next: Continuous case and seismic
Up: Interpolation with Fourier basis
Previous: Continuous Fourier basis
Stanford Exploration Project
11/11/1997