A generalized version of the 2-microlocal frontier prescription

TL;DR

This paper extends the theory of 2-microlocal frontiers by unifying previous results and characterizing a broad class of functions with prescribed local regularity profiles, including line cases.

Contribution

It generalizes existing results by providing a unified framework for constructing functions with a specified 2-microlocal frontier, including a complete characterization for linear frontiers.

Findings

Unified class of functions with prescribed 2-microlocal frontiers

Constructed functions for any given monotone concave downward curve

Characterized functions with linear 2-microlocal frontiers

Abstract

The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $\mathbb {R}^2$, provides a complete and profound classification of pointwise singularity. In \cite{Meyer1998}, \cite{GuiJaffardLevy1998} and \cite{LevySeuret2004} the authors show the following: given a monotone concave downward curve in the plane it is possible to exhibit one function (or distribution) such that its 2-microlocal frontier al $x_0$ is the given curve. In this work we are able to unify the previous results, by obtaining a large class of functions (or distributions), that includes the three examples mentioned above, for which the 2-microlocal frontier is the given curve. The three examples above are in this class. Further, if the curve is a line, we characterize all the functions whose 2-microlocal frontier at $x_0$ is the given line.