Let 
be a nonempty set, then a filter on
is a nonempty collection 
of subsets of 
having the following properties:
1. 
,
2. If 
, then 
,
3. If 
and 
then 
If 
is an infinite set, then the collection
is a
filter called the cofinite (or Fréchet)
filter on 
.
In signal processing, a filter is a function or procedure which removes unwanted parts of a signal. The concept of filtering and filter functions is particularly useful in engineering. One particularly elegant method of filtering Fourier transforms a signal into frequency space, performs the filtering operation there, then transforms back into the original space (Press et al. 1992).
