Naturally we may prefer true dip filters, that is, functions of instead of the functions of described above. But it can be shown that replacing in the above expressions by gives recursions that are unstable.
Sharper pie slices (filters which are more strictly a rectangle function of ), may be defined through a variety of approximation methods described by Hale and Claerbout . Generally, |k| can be expanded in a power series in .If the approximation to |k| is ensured positive, you can expect stability of the recursion that represents .
More simply, you might be willing to Fourier transform time or space, but not both. In the remaining dimension (the one not transformed) the required operation is a highpass or lowpass filter. This is readily implemented by a variety of techniques, such as the Butterworth filter.