Apodization and windowing eigenfunctions.

Across a range of spectral estimation problems and beam focusing problems, it is necessary to constrain the properties of a function and its Fourier transform. In many cases, compact functions in both domains are desired, within the theoretical bounds of the uncertainty principle. Recently, a hyperbolic sine function of modified argument and power was found to be an approximate eigenfunction of the Fourier transform operation, and demonstrated useful properties of compactness with low side lobes. The empirical finding of the eigenfunction relationship is explained by comparison with the prolate spheroidal wave functions, which have exact eigenfunction properties, and their usefulness is demonstrated by examples.