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).