Coxeter system of lines are sets of injectivity for the twisted spherical means on $\mathbb C$

TL;DR

This paper establishes that lines through the origin and Coxeter systems of lines are sets of injectivity for twisted spherical means on complex plane functions with specific spectral properties, extending known results from real to complex settings.

Contribution

It proves that all lines through the origin are injectivity sets for twisted spherical means on certain complex functions, and that Coxeter systems of lines are injectivity sets for a broader class of functions.

Findings

Lines through the origin are injectivity sets for TSM on specific complex functions.

Coxeter systems of even lines are injectivity sets for TSM on $L^q$ functions.

Extension of real line injectivity results to complex plane with spectral conditions.

Abstract

It is well known that a line in $\mathbb R^2$ is not a set of injectivity for the spherical means for odd functions about that line. We prove that any line passing through the origin is a set of injectivity for the twisted spherical means (TSM) for functions $f\in L^2(\mathbb C),$ whose each of spectral projection $ e^{\frac{1}{4}|z|^2}f\times\varphi_k$ is a polynomial. Then, we prove that any Coxeter system of even number of lines is a set of injectivity for the TSM for $L^q(\mathbb C),~1\leq q\leq2.$