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:
where square brackets, 
, indicate the order of the operators and
round brackets, 
, indicate the arguments of the input and output
functions. Without loss of generality we consider here functions
with zero mean value. Note that because of the finite and infinite
correlation length of stochastic and periodic series, respectively,
no unique normalization C applies in the continuous case.
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.
(
) 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:
where 
denotes the Dirac symbol: 
.
In the discrete case, 
assumes the value 
for x and 0
elsewhere.
