LULU operators and locally monotone approximations

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