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
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 .
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.