Real analytic expansion of spectral projection and extension of Hecke-Bochner identity

TL;DR

This paper extends Hecke-Bochner identities for spectral projections in complex spaces using Weyl correspondence and proves spheres are injective for twisted spherical means, providing a real analytic expansion of spectral projections.

Contribution

It introduces an extension of Hecke-Bochner identities for spectral projections and establishes spheres as injective sets for twisted spherical means with real analytic weights.

Findings

Extended Hecke-Bochner identities for spectral projections.

Proved spheres are injective for twisted spherical means.

Derived a real analytic expansion for spectral projections.

Abstract

In this article, we review the Weyl correspondence of bigraded spherical harmonics and use it to extend the Hecke-Bochner identities for the spectral projections $f\times\varphi_k^{n-1}$ for function $f\in L^p(\mathbb C^n)$ with $1\leq p\leq\infty.$ We prove that spheres are sets of injectivity for the twisted spherical means with real analytic weight. Then, we derive a real analytic expansion for the spectral projections $f\times\varphi_k^{n-1}$ for function $f\in L^2(\mathbb C^n).$