Transformations which take functions, e.g. x, y as arguments and
return functions as results are called operators. The direct and
inverse Fourier transform,
and the convolution,
*, are operators defined in the following way:
The discrete operators and * are well defined only for observations and frequencies which are spaced evenly by and , respectively, and span ranges and . Then and only then reduces to orthogonal matrices. It follows directly from Eq. (12.2) that we implicitly assume that the observations and their transforms are periodic with the periods and , respectively. The assumption is of consequence only for data strings which are short compared to the investigated periods or coherence lengths or for a sampling which is coarse compared to these two quantities. Such situations should be avoided also in the general case of unevenly sampled observations.
The following properties of
and * are