Slicing pies

Naturally we may prefer true dip filters, that is, functions of $k / \omega$ instead of the functions of $k^2 / \omega$ described above. But it can be shown that replacing $k^2 / \omega$ in the above expressions by $k^2 / \omega^2$ gives recursions that are unstable.

Sharper pie slices (filters which are more strictly a rectangle function of $k / \omega$), may be defined through a variety of approximation methods described by Hale and Claerbout [1983]. Generally, |k| can be expanded in a power series in $\partial^2 / \partial x^2$.If the approximation to |k| is ensured positive, you can expect stability of the recursion that represents $\vert k\vert/ i \omega$.

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.