Analysis of directional higher order jump discontinuities with trigonometric shearlets

TL;DR

This paper extends the analysis of trigonometric shearlets to detect higher order directional jump discontinuities in multivariate periodic functions, providing bounds for shearlet coefficients related to these features.

Contribution

It generalizes previous results to higher order derivatives and develops bounds for shearlet coefficients in this more complex setting.

Findings

Shearlets can detect higher order directional discontinuities.

Derived bounds for shearlet coefficients related to jump discontinuities.

Extended localization and decay properties of shearlet inner products.

Abstract

In a recent article, we showed that trigonometric shearlets are able to detect directional step discontinuities along edges of periodic characteristic functions. In this paper, we extend these results to multivariate periodic functions which have jump discontinuities in higher order directional derivatives along edges. In order to prove suitable upper and lower bounds for the shearlet coefficients, we need to generalize the results about localization- and orientation-dependent decay properties of the corresponding inner products of trigonometric shearlets and the underlying periodic functions.