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
noteworthy: