Wave fronts via Fourier series coefficients

TL;DR

This paper introduces a novel method to describe the wave front and Sobolev-type wave front of distributions using Fourier series coefficients, providing new insights into the microlocal analysis of distributions.

Contribution

It offers a new characterization of wave fronts for distributions based on Fourier series coefficients, advancing the understanding of their microlocal properties.

Findings

New Fourier series coefficient-based description of wave front

Enhanced understanding of microlocal analysis of distributions

Potential applications in analyzing periodic distributions

Abstract

Motivated by the product of periodic distributions, we give a new description of the wave front and the Sobolev-type wave front of a distribution $f\in\mathscr{D}'(\mathbb{R}^d)$ in terms of Fourier series coefficients.