The Lu Lu-semigroup for Envelopes Of Functions

Abstract

In the theory of non-linear smoothers a semigroup of operators has been established, that structures the rank based selectors in a lattice for the purpose of comparison and analysis. The well known upper and lower envelopes of functions can be associated with a pair of operators L and U that allow a similar structure to be defined. The monoid of operators that are compositions of these is established, and the inequalities needed for partially ordering are proved. Some interpretations, analogies and uses are discussed.