LULU operators for functions of continuous argument

TL;DR

This paper extends LULU operators from sequences to continuous functions, maintaining key properties like local trend preservation and total variation, thus broadening their applicability in nonlinear analysis.

Contribution

It introduces a continuous domain version of LULU operators that retains the core properties of the discrete operators, enabling new applications in continuous function analysis.

Findings

Operators form a fully ordered semi-group of four elements.

They preserve local trend.

They maintain total variation.

Abstract

The LULU operators, well known in the nonlinear multiresolution analysis of sequences, are extended to functions defined on a continuous domain, namely, a real interval. We show that the extended operators replicate the essential properties of their discrete counterparts. More precisely, they form a fully ordered semi-group of four elements, preserve the local trend and the total variation.