Discrete Fourier basis

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  

$$\psi_k (x) = \frac{1}{\sqrt{2N}} e^{i \pi \frac{k}{N} x} \;,$$ (48)

where the discrete frequency index k also ranges, according to the Nyquist sampling criterion, from -N to N. The interpolation function is computed from equation () to be

$$\begin{eqnarray} W (x, n) & = & \frac{1}{2 N} \sum_{k=-N}^{N-1} e^{i \pi \frac{... ... \left[\pi (x - n)\right]}{2N \sin\left[\pi (x - n)/2N\right]} \;.\end{eqnarray}$$ (49)

An interpolation function equivalent to () has been found by Muir Lin et al. (1993); Popovici et al. (1993, 1996). It can be considered a tapered version of the sinc interpolant () with smooth tapering function

$$\frac{\pi (x - n)/2N}{\tan\left[\pi (x - n)/2N\right]}\;.$$

Unlike most other tapered-sinc interpolants, Muir's interpolant () satisfies not only the obvious property (), but also properties () and (), 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 .

 

ma-sinc
Figure 11 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 interpolant. The top is for N=2 (5-point interpolation); the bottom, N=6 (13-point 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 would go beyond the scope of the present work.

The next section carries the analysis to the continuum and compares the mathematical interpolation theory with the theory of seismic imaging.