On the Titchmarsh convolution theorem for distributions on the circle

TL;DR

This paper extends the Titchmarsh convolution theorem to distributions on the circle, revealing that violations occur only under specific symmetry conditions, thus refining understanding of convolution properties in this setting.

Contribution

It establishes a version of the Titchmarsh theorem for circle distributions and identifies symmetry conditions necessary for violations.

Findings

Violations of the naive Titchmarsh theorem are possible under certain symmetry conditions.

The theorem holds generally except when both distributions have specific symmetries.

Provides a refined understanding of convolution behavior on the circle.

Abstract

We prove a version of the Titchmarsh convolution theorem for distributions on the circle. We show that the "naive form" of the Titchmarsh theorem could be violated, but that such a violation is only possible for the convolution of distributions which both possess certain symmetry properties.