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

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 $e^{\frac{1}{4}|z{|}^2}f\times {\phi }_k$ is a polynomial. Then, we prove that any Coxeter system of even number of lines is a set of injectivity for the TSM for $L^q(\mathbb{C}),1\le q\le 2.$

This article is accepted in J. Funct. Anal. jointly with arXiv:1204.2773

References in corpus (1)

• On the injectivity of the circular Radon transform arising in thermoacoustic tomography

Cited by in corpus (3)

• Real analytic expansion of spectral projection and extension of Hecke-Bochner identity
• Inversion Formulas for the Spherical Means in Constant Curvature Spaces
• Non-harmonic cones are sets of injectivity for the twisted spherical means on ${\mathbb{C}}^n$