PROJECTIONS AND SEPARATORS

Abstract

Projections are important in many fields of analysis, specifically, in approximation theory and signal analysis. They can be perceived to separate a vector (function, sequence) into a component in a chosen specific subspace and the residue, or the “signal” and “noise”. In the framework of linear theory a projection is idem-potent and an idempotent linear operator is a projection. When an operator is not linear the ideas are still partially applicable and useful. A simple property, complementary to idempotence, extends the analogy in a useful way. This is particularly so in the theory of non-linear smoothers, when selectors are analysed and compared in a LULU-structure.