LULU operators and locally monotone approximations

Abstract

The LULU operators, well known in the nonlinear multiresolution analysis of sequences, are extended to functions defined on continuous domain, namely, a real interval $Ω\subseteq\mathbb{R}$. Similar to their discrete counterparts, for a given $δ>0$ the operators $L_δ$ and $U_δ$ form a fully ordered semi-group of four elements. It is shown that the compositions $L_δ\circ U_δ$ and $U_δ\circ L_δ$ provide locally $δ$-monotone approximations for the bounded real functions defined on $Ω$. The error of approximation is estimated in terms of the modulus of nonmonotonicity.